Silence of Scripture
This has to do with what we call the "silence" of scripture — things that are not explicitly discussed — and whether silence is permissive or prohibitive. In the general case (not just scripture), we often think of silence as permissive. However, it is easy to show that this is not always true. I'll provide an example:
Let's say you send your teenage child to the store for milk and eggs, and give him/her a $20 bill to cover the expense. The child comes back with milk, eggs, ice cream, and less change than you hoped for. You didn't tell him/her not to buy the ice cream, but you're probably not going to be happy. Here, the silence was prohibitive. You also didn't specify which store to go to. You may have expected the local grocery store, but if the child knew of a sale at a nearer convenience store they may have chosen that instead. Here, the silence was permissive.
Applied to scripture, there is a common (not universal) interpretation that when we are given instructions in an area of worship, silence on other aspects of the area is prohibitive. For example, God told Noah to use gopher wood when building the ark. He didn't mention other kinds of wood (he was silent), but his specific instruction for gopher wood excludes/prohibited them. What if Noah wanted to use gopher wood for the hull, but thought a little bit of maple trim might have looked nice around the window and door? This interpretation system would not allow for that. Going back to the shopping example, ice cream was not on the list, and so was prohibited.
The above system is an example of a hermeneutic. Hermeneutics are the lens you use to study scripture. As an example, they are how you know what to do when you come upon "Do not judge, or you too will be judged." in one place, and "Do you not know that the Lord’s people will judge the world?" in another. Hermeneutics help ensure that you interpret scripture consistently; that you not hear only what you want to hear from the passages you want to read. A perfect hermeneutic, perfectly applied, would lead to a perfect understanding of scripture. Unfortunately, man is far from perfect.
Most good hermeneutics will at some point need to deal with the silence of scripture, in some way or another. Most fail to handle it adequately. The hermeneutic described above actually handles the subject pretty well, but hermeneutics are a creation of man, and so are not flawless or perfect. Our application of them also often leaves something to be desired. In spite of that, if you don't have a hermeneutic system to use when you study scripture, you're probably not getting what you should out of your study. At the same time, it's important to remember that rather than hard-fast rules they should be seen more as guidelines.
Answering the Title Question
This specific hermeneutic is normally only used to apply to worship. Silence of scripture is more often taken to be permissive in the general sense, but for worship specifically it might be a good idea to pay more attention to what is and what is not enumerated explicitly. Therefore, with regards to the title question, the answer is most often permissive. However, it's still important to make sure that whatever you do, you're making choices that are centered on God, and not choices that are centered on what you can get away with.
To fully understand the instrumental music in the churches of Christ, you need to know something of our history. The churches of Christ descend from the Restoration movement of the early/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 the 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 promote hermeneutics that would 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... the Restoration Movement was 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 American group in existence at the time to make it through the Civil War without dividing over the issue.
churches of Christ
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.
Silence of Scripture
The first is a very strict interpretation on the silence of scripture, as described in the section above. The upshot is that churches of Christ believe we are lacking an explicit instruction to use instruments (while singing or otherwise) in congregational worship — that scripture is silent here. We discount the passages in the Psalms quoted in other answers as allegorical, specific to Old Testament worship, or applying to personal life rather than congregational worship. Churches of Christ acknowledge that 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.
To put it another way, churches of Christ don't use incense during worship either, and you likely don't think this is strange — in fact, you would likely feel a little odd to see/smell it — after all, God does not instruct us anywhere that he wants incense as part of our worship. He's silent about incense, and so we choose to focus on the areas where He is more direct in what He wants. This argument is weak when left by itself, 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. Additionally, 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, 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.
The third 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 (sorry, no reference handy :( ) 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 any historical examples of any instrumental accompaniment. 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.
One other reason is that as a capella music for worship became less and less common in the culture, retaining and featuring a capella became part of the identity of the churches of Christ. Identity is especially important for a group as loosely associated as the churches of Christ, who recognize no earthly central authority or association. This final piece is by far the weakest from a persuasive or justification standpoint, but it still needs to be mentioned.
Put the four together, and you have a fairly strong case that a cappella music is the way to go in the worship service. Unfortunately, more recent history in churches of Christ have forgotten that the reasoning here is as much practical as it is scriptural (especially with regards to the fourth reason, though, again, the scriptural argument is not entirely without merit) leading to a stronger reaction against instrumental music from some contingents within the churches of Christ than may be warranted. Remember, again, that this is a group with no earthly central authority or governing body, and therefore anything you hear (including this very text) is the opinion of the speaker; never of the churches of Christ as a whole.
There are also indications that you do not need to avoid instrumental music in personal and corporate worship. For example, the historical evidence is that the early worship followed the Jewish synagogue model. Jewish synagogue worship was entirely a capella, but they were happy to use instruments in their personal and family worship as well as the temple/corporate setting.
In other words, there is something to this silence of Scripture thing, but it's not fool proof. There's also something to be said for using a capella music during the formal church service. But you're probably in safe water to listen to a little Jars of Clay, Newsboys, or Reliant K with your friends or on your own time. The important thing is the make sure you are tuning into what God wants here rather than what you want.