Women should remain silent in the churches. They are not allowed to speak, but must be in submission, as the law says. (TNIV)T.L. raises five objections to a strictly literal interpretation of this verse that women must never say anything in church. Here is the fifth objection:
However, the rest of verse 34 is confusing, because clearly women are permitted by Paul to speak, there is no obvious relationship to speaking and being submissive, and there is no Scriptural Law that addresses this issue. It is possible that there is a local cultural law, but would seem unlikely for Paul to site it. However, it is possible that he is referencing some local laws relative to the respect of women toward their husbands, of which there were numerous, both by the Greeks and Jews. Honor and Shame in the World of the Bible by V. H. Matthews et al, is supposed to discuss many of these.
There are two other possible interpretations being considered by present day theologians. A. C. Thiselton (in his new NIGTC)  suggests that the questions could be regarding sifting and weighing the words of the prophets, i.e. judging the prophets. While this is a good possibility, I think one needs to remember that the general congregations are not the ones to judge the words of the prophets either, but the prophets as a group, possibly including other anointed leadership.
Another possible interpretation is that verses 34 and 35 are another¹s words that Paul is quoting and actually repudiating in verse 36. Or did the word of God come originally from you? Or was it you only that it reached? Paul has quoted others and then responded to them in other places. I believe this is called an interpolation and that Fee argues this in First Epistle. 
The first makes sense since women were coming out of a place of ignorance and most needed to learn more before they could offer informed constructive discussion. The second makes sense if Paul is addressing those who would think only men would ever be capable of discernment since it is the Holy Spirit who inspires and brings truth.
What do you think? Was Paul refuting an extrabiblical "law" against women speaking in church? If not, what kind of silence did he want from women, given that he had already written about other kinds of women's speech in church (e.g. 1 Cor. 11:5)? Was Paul addressing a local church problem only or stating a rule for women's speech in church services for all time?