I'm familiar with a couple different schools of thought on this. Doubtless there are others as well.
The first comes from Greek Orthodox tradition. I can't really speak to it directly, but a quick google search turned up a result indicating that it might have more to do with history and wanting to separate themselves from pagan worship. But there's likely a lot more to it.
The other is from the churches of Christ. This side I can speak to at length. To understand this, you need to know something of their history. The churches of Christ descend from the Restoration movement of the mid-1800s. This was a movement that strove to throw off all of the excess structure that had arisen over the years since the 1st century and restore 1st century-style worship. The primary means of achieving this goal was through a very careful and strict interpretation of scripture. "Innovations" from known historical patterns of the first century should only be permitted when there is a clear case for them, shown either through direct command, example, or necessary inference.
The reason for the return to 1st century worship was a reaction to all divisions, and a desire to fulfill the Lord's prayer "that they may all be one." It was a drive for unity, and one part of this was to push away anything that a true seeker might possibly see as contrary to scripture, and hold up only those things that meet an exceptionally strict criteria for doctrinal purity (the other two parts are congregational independance/freedom and an strong emphasis on individual study). Don't knock the process too hard, as it met with some success... this is one of the few major groups to begin it's history as the result of a merger of a few smaller groups, rather than as a split from a larger group, and was the only group in existence at the time to make it through the Civil War without dividing over the issue (sadly, I can't find a reference for this any more).
There are four pieces that combine to contribute the general (not universal, but nearly so) use of a cappella music among church of Christ congregations.
The first is a very strict interpretation on the silence of scripture. I don't want to go into this at length, but you can read my answer to this question on faith healing to get a vague idea on how it works. The upshot is that we are lacking explicit New Testament instructions or examples on the use of instruments (while singing or otherwise) in congregational worship — scripture is silent here, but we do have explicit instructions to sing. Because we have an explicit instruction about how to use music in worship, the silence in this case is prohibitive rather than permissive. This by itself is weak, but remember that it's only part of the reasoning.
The second reason is participatory. The command to sing praises is interpreted to mean that music in worship should include everyone as a participant... that if the music portion of your service consists mainly of just a few performers or even a choir, while the majority of the congregation is only listening (even if this is the effect rather than the intent), you're doing it wrong. The entire congregation should take part in producing the music. Instruments are seen as counter-productive to this effort, as history shows they have a tendency to take over and dominate the performance to the exclusion, rather than inclusion, of audience participation.
A third reason is practical. Successful use of a cappella music in worship is something that takes a commitment. If you decide to just try out a cappella music one week in service, or just do one a cappella song in each service, you're likely to fail miserably if few of the participants have done this before. Therefore, as a practical matter the choice to use a cappella music came down to an "all or nothing" approach, where those that use a cappella music in worship tend to do so exclusively. If you see a commandment to sing as a congregation as something to take seriously, a commitment to an a cappella worship service is something to look at.
The final reason is historical. Remember that this group comes out of a movement whose goals were to restore 1st century worship styles. We do know from historical documents that the earliest Christians, without exception, did not use instruments in their worship. At all. In fact, it's already the third century before we find examples of any instrumental accompaniment.1 It's possible there were forces other than doctrine driving this: for example, funds, fear of persecution if louder instruments gave away a secret service, or simply modeling congregational worship after Jewish synagogue worship (which also did not use instruments). But the fact of the matter is that the historical record indicates a cappella is the way to go.
1 Not my original source, but I recently came across this article that has more info on the historical sources, with lots of research references: http://www.housetohouse.com/HTHPubPage.aspx?pub=2&issue=663