There are several arguments used against a straight interpretation of 1Cor 11. First, that it was cultural command which we can ignore, second, that the covering is simply hair, and third, that the passage actually refers to having hair 'pinned up' not 'covered'. I will go through each of the arguments.
The first argument given is that the whole head-covering instruction was cultural (something about the prostitutes having shaved heads), and it no longer applies to us. The problem with this argument is that Paul's reasoning was timeless -- he discusses Headship, Glory, Creation, Angels, Nature, and then rounds it off by saying "But if one is inclined to be contentious, we have no other practice, nor have the churches of God." (1Cor 11:16) This issue wasn't cultural, it was eternal, and modern men and women must have a clear understanding of what this passage is commanding us.
The simplest argument against cloth headcoverings reasons that verse 15 ("...For her hair is given to her for a covering.") indicates that the covering discussed in the preceding verses is a woman's hair. The problem lies in verses 5 & 6
But every woman who has her head uncovered while praying or prophesying disgraces her head, for she is one and the same as the woman whose head is shaved. For if a woman does not cover her head, let her also have her hair cut off; but if it is disgraceful for a woman to have her hair cut off or her head shaved, let her cover her head. 1Cor 11:5-6You see, if the covering is really just long hair, then being "uncovered" would mean having short hair. Yet, how can she cut off her hair if it's already short? These verses simply cannot be reconciled with the interpretation that a woman's hair is her only covering. Aslo, the whole passage seems far too complicated if all Paul meant was women should have long hair, and men should have short hair. The passage is either saying what it means to say (that women should wear a cloth head covering and men should not), or it is mistranslated.
One argument I read stated that the first word Paul used for "cover" was properly translated as "long hair pinned up", and that the actual word properly translated as "cover" isn't used until v15. The same argument contends that the word often translated "as" or "for" in that same verse is properly translated as "instead of", so that the actual verse reads "For her hair is given to her instead of a (cloth) covering".
This sounded like the most convincing argument I had heard against head coverings, if in fact the translations were incorrect. Therefore, I decided to delve into Strong's Concordance, Thayer's Lexicon, and the Septuagint to get at the root of the words. Now, more than ever, I am convinced that the way a woman is to cover her head during prayer and prophesy is with a cloth veil which hides her hair. Following is the process I took to reach my conclusion. Keep in mind that although I have no expertise in languages (I can't even learn French), if you treat words like code, you can get a pretty decent understanding of their appropriate use.
Following is the passage with Strong's word numbers next to each word. The words pertinent to this discussion are in bold. I would encourage you to take out your favourite translation (no paraphrases, please), and write down Strong's word number next to the bold selections. This might help you connect the definitions to the appropriate words.
1Cr 11:3 But  I would have  (5719) you  know  (5760), that  the head  of every  man  is  (5748) Christ ; and  the head  of the woman  [is] the man ; and  the head  of Christ  [is] God .First, let's take a look at when this directive is in place.
1Cr 11:4 Every  man  praying  (5740) or  prophesying  (5723), having  (5723) [his] head  covered , dishonoureth  (5719) his  head .
1Cr 11:5 But  every  woman  that prayeth  (5740) or  prophesieth  (5723) with [her] head  uncovered  dishonoureth  (5719) her  head : for  that is  (5748) even  all one  as if  she were shaven  (5772).
1Cr 11:6 For  if  the woman  be   not  covered  (5743), let her   also  be shorn  (5669): but  if  it be a shame  for a woman  to be shorn  (5670) or  shaven  (5745), let her be covered  (5744).
1Cr 11:7 For  a man  indeed  ought  (5719) not  to cover  (5745) [his] head , forasmuch as he is  (5723) the image  and  glory  of God : but  the woman  is  (5748) the glory  of the man .
1Cr 11:8 For  the man  is  (5748) not  of  the woman ; but  the woman  of  the man .
1Cr 11:9   Neither  was   the man  created  (5681) for  the woman ; but  the woman  for  the man .
1Cr 11:10 For this  cause  ought  (5719) the woman  to have  (5721) power  on  [her] head  because  of the angels .
1Cr 11:11 Nevertheless  neither  is the man  without  the woman , neither  the woman  without  the man , in  the Lord .
1Cr 11:12 For  as  the woman  [is] of  the man , even so  [is] the man  also  by  the woman ; but  all things  of  God .
1Cr 11:13 Judge  (5657) in  yourselves  : is it  (5748) comely  (5723) that a woman  pray  (5738) unto God  uncovered ?
1Cr 11:14 Doth  not even  nature  itself  teach  (5719) you , that , if   a man  have long hair  (5725), it is  (5748) a shame  unto him ?
1Cr 11:15 But  if  a woman  have long hair  (5725), it is  (5748) a glory  to her : for  [her] hair  is given  (5769) her  for  a covering .
1Cr 11:16 But  if any man  seem  (5719) to be  (5750) contentious , we  have  (5719) no  such  custom , neither  the churches  of God .
Praying  proseuchomai pros-yoo'-khom-ahee from 4314 and 2172; to pray to God, i.e. supplicate, worship:--pray (X earnestly, for), make prayer.
Prophesying  propheteuo prof-ate-yoo'-o from 4396; to foretell events, divine, speak under inspiration, exercise the prophetic office:--prophesy.
Chapter 11 is often prefaced with the title "Directions for Public Worship", however, the instructions for the covering/uncovering of one's head do not specify whether the praying or prohesying are on the Lord's Day, in public worship. In the preceeding chapter, Paul discusses the Lord's Supper, and then broadens the topic to the eating of sacrificed meat (outside of public worship). Paul then thanks the church for holding fast to the traditions as he delivered them, and goes on to address head coverings, before veering back to the Lord's Supper. At the very least, one may conclude that head coverings apply to Lord's Day assembly (the "traditions"), however, an argument could easily be made that the instructions apply to all praying and prophesying, regardless of time or location.
Second, let's see who we are dealing with.
Man  aner an'-ayr a primary word (compare 444); a man (properly as an individual male):--fellow, husband, man, sir.
Woman  gune goo-nay' probably from the base of 1096; a woman; specially, a wife:--wife, woman.
"Gune ", according to Thayer's Lexicon, can mean "1. a woman of any age, whether a virgin, or married, or a widow", or specifically, "2. a wife, or a betrothed woman". Therefore, at the very least, Paul is referring to married or betrothed women. This interpretation makes sense in light of Paul's argument concerning headship. Not every man is the head of every woman, but rather a Husband is the head of his own Wife.
Third, let's look at the various words used for "cover" and "uncover". The first word "kata " is only used by itself in v4 in relation to men. Afterwards, it is used in combination with "kalupto  to form "katakalupto " and "akatakaluptos ". Notice that when hair is referred to as a "covering", a completely different word (peribolaion) is used. I believe this means that although hair is given as a type of covering, it is not the covering required while praying or prohesying. For reference, I have included the definitions for "shaved" and "shorn", as well as "hair" and "long hair".
 kata kat-ah' a primary particle; (prepositionally) down (in place or time), in varied relations (according to the case (genitive, dative or accusative) with which it is joined):--about, according as (to), after, against, (when they were) X alone, among, and, X apart, (even, like) as (concerning, pertaining to touching), X aside, at, before, beyond, by, to the charge of, (charita-)bly, concerning, + covered, (dai-)ly, down, every, (+ far more) exceeding, X more excellent, for, from ... to, godly, in(-asmuch, divers, every, -to, respect of), ... by, after the manner of, + by any means, beyond (out of) measure, X mightily, more, X natural, of (up-)on (X part), out (of every), over against, (+ your) X own, + particularly, so, through(-oughout, -oughout every), thus, (un-)to(-gether, -ward), X uttermost, where(-by), with. In composition it retains many of these applications, and frequently denotes opposition, distribution, or intensity.
1) down from, through out
2) according to, toward, along
 kalupto kal-oop'-to akin to 2813 and 2928; to cover up (literally or figuratively):--cover, hide.
1) to hide, veil
a) to hinder the knowledge of a thing
 (5743) katakalupto kat-ak-al-oop'-to from 2596 and 2572; to cover wholly, i.e. veil:--cover, hide.
1) to cover up
2) to veil or cover one's self
 akatakaluptos ak-at-ak-al'-oop-tos from 1 (as a negative particle) and a derivative of a compound of 2596 and 2572; unveiled:--uncovered.
1) not covered, unveiled
 peribolaion per-ib-ol'-ah-yon neuter of a presumed derivative of 4016; something thrown around one, i.e. a mantle, veil:--covering, vesture.
1) a covering thrown around, a wrapper
a) a mantle
b) a veil
 xurao xoo-rah'-o from a derivative of the same as 3586 (meaning a razor); to shave or "shear" the hair:--shave.
1) to shear, shave
2) to get one's self shaved
 keiro ki'-ro a primary verb; to shear:--shear(-er).
1) to sheer: a sheep
2) to get or let be shorn
3) of shearing or cutting short the hair of the head
 komao kom-ah'-o from 2864; to wear tresses of hair:--have long hair.
1) to let the hair grow, have long hair
 kome kom'-ay apparently from the same as 2865; the hair of the head (locks, as ornamental, and thus differing from 2359; which properly denotes merely the scalp):--hair.
1) hair, head of hair
The following definitions are taken from Numbers 5:18. The Septuagint uses "apokalupto " (a derivitive of "kalupto ") to translate "para ". Some arguments against head coverings use this verse to prove that "katakalupto " means "long hair pinned up", and therefore "akatakalupto " means "long hair let down", or something to that effect. However, by examining the use of the Hebrew "para " one understands that is not the case. Although "para " does mean "to let loose", it appears to have the connotations of uncovering and nakedness, rather than of disheveled hair.
 apokalupto ap-ok-al-oop'-to from 575 and 2572; to take off the cover, i.e. disclose:--reveal.
 para paw-rah' a primitive root; to loosen; by implication, to expose, dismiss; figuratively, absolve, begin:--avenge, avoid, bare, go back, let, (make) naked, set at nought, perish, refuse, uncover.
2) to let go, let loose, ignore, let alone
1) to let go, let loose
2) to let alone, avoid, neglect
3) to loosen
b) (Niphal) to be let loose, be loosened of restraint
1) to cause to refrain
2) to show lack of restraint
3) to let loose restraints
Fourth, let's look at the definition of "anti " in v16.
 anti an-tee' a primary particle; opposite, i.e. instead or because of (rarely in addition to):--for, in the room of. Often used in composition to denote contrast, requital, substitution, correspondence, etc.
1) over against, opposite to, before
2) for, instead of, in place of (something)
a) instead of
c) for that, because
d) wherefore, for this cause
Although "anti " can be translated as "instead of", most translations use "for" or "as".
Finally, let's examine the implications of saying the covering is "long hair pinned up". If you are using this intepretation of the text to avoid wearing a cloth head covering, do you wear your hair long and pinned up? If not, then the text says you should shave your head, for doing so is disgraceful. However, since it is a disgrace to have a shaved head, you should wear a cloth veil to cover your shame.
If you are preaching this interpretation, then you must also be teaching women to a) wear their hair long; and b) pin their hair up; otherwise, c) they must cover their short or long, unpinned hair with a cloth veil.
It is interesting to note that of all the depictions of female believers throughout the ages, not one is shown with "long hair pinned up", but rather they are shown wearing a cloth head covering. It appears that even Paul's first audience took his words as instructions to wear a cloth veil.
To recap, at the very least, you should be covering your hair with a cloth veil or scarf if you are a believing married woman praying/prophesying in public worship on the Lord's Day. I would encourage believing married women to keep the cover on during the entire service (otherwise it becomes an unnecessary distraction). I believe it would be highly beneficial to wear the same head covering during group bible studies and prayer meetings, as well as private study and prayer. I believe Paul's argument for wearing a head covering applies to prayer and prophesy every day, not just on Sunday. Also, there is nothing wrong or sinful with wearing a head covering at all times. I believe this practice blesses sporadic, spontaneous prayer. It could also be beneficial to women struggling in areas of Godly submission towards their Husbands and God.
As I mentioned earlier, there are several eteranl reasons Pauls gives for headcoverings: Headship, Glory, Creation, Angels, Nature, and then rounds it off by saying "But if one is inclined to be contentious, we have no other practice, nor have the churches of God." (1Cor 11:16). I believe the simplest reason he gives has to do with Glory, so let's work through that chain: The glory of God is man (v7), the glory of man is woman (v7), and the glory of woman is her hair (v15). When we're worshiping God, who's glory are we exalting? God's or our own? Since we are exalting God, we should be uncovering the glory of God and covering the glory of man and woman. Since God's glory is man, he should be uncovered. Since man's glory is woman, she should be covered. Since woman's glory is her hair, it should be covered. Therefore, we have two reason here to cover our heads and our hair -- to cover man's glory as well as our own. You may disagree with the idea that nature teaches that long hair on men and short hair on women is shameful, and you might not comprehend what angels have to do with signs of authority, but the notion of uncovering God's glory and covering man's should be easily understood. Afterall, isn't that what worship is all about?
As important as the headcovering may be, please keep in mind that souls are far more important than any garment a person may or may not be wearing. Before taking a woman to task for not wearing a proper covering, or a man for wearing an improper covering, it is essential that you determine that they are indeed a believer in the Lord Jesus Christ. God sees beyond the external, into the heart. A woman can wear a headcovering at all times, but if her sins were not paid for on the Cross, then she is little more than a white-washed tomb. This principle should be applied to all areas of life. There is no sense in dealing with anyone's sins unless they are in a saving relationship with Jesus.
Headcovering in Public Worship (Pastor Brian Schwertley, sermon)
Headcoverings II (Pastor Brian Schwertley, sermon)
Sermons on Headcoverings (all sides)