Some Christians will listen to music composed only by professed Christians. This is, I believe, fundamentally misguided.
Here's why: There is an apparent assumption at work here and it is that the religious profession of the musician is paramount when considering music. It is not. One may be a fervent Christian and write music of poor quality. One may not profess faith and write excellent music. Just as truth is not person relative, so aesthetic excellence is not person relative. Mozart, for example (or the great Beethoven, as Vic noted), was not a Christian but, from any objective musical analysis, wrote remarkable music.
Many Christians at this point may frown and protest, "All musical analyses are purely subjective, based merely on personal taste." There is certainly a signficant aspect of taste to music, in terms of what particular persons may like or dislike. Such a subjective aspect, however, does not mean that all is pure subjectivity. Music is both an art and a science to which various sorts of criticism may be brought: some music, across all musical spectra, well expresses all the elements of musical composition and may be said to be beautiful while some music fails, even within its genre, to do so and is rightly deemed inferior.
It's interesting when Christians demur here and insist that all is a matter of taste. Christians would never be relativists in metaphysics, epistemology, or ethics, but, for some reason, find relativism warranted in aesthetics. No, aesthetics has standards too, as do all arts, crafts, and all human endeavors. The principles for such are found in Scripture (we are to endeavor to do all things well, to the glory of God) while, in each case, the particulars reside within the craft itself, as a good creation of God. In other words, because the Bible does not give details for plumbing (an oft used illustration these days), this does not mean that such do not reside in and are developed from the craft. The Bible has more to say about music than many tasks but not compositionally, as such. It's simply ignorance about music as a craft that prompts one to insist that there are no standards. No real musician thinks that.
All this, then, has been to say that the real question is not whether music is written by a Christian, but whether the music evinces the excellence that it is meant to as a good gift of our gracious God. Much "Christian" music does not. Much "non-Christian" music does. I listen to a lot of great music written by professing Christians and others. Some of this is contemporary. I am not arguing that all good music is old. But all good music is good music (there's some music that is objectively good that I care less for than other good music).
I am about to give a lecture to a group of potential future ministers on the topic, "The Minister and Music," which I've developed under these three points: "Learn the psalter and the hymnal" (using them devotionally as well as liturgically); "Look for help from the musicians in your church" (in terms of helping to sing the church's music and be aware of performance practices); and, lastly, "Listen to good music" (and here I will cite, among others, Bach, Handel, Haydn, Brahms, Vaughn Williams, and Rutter). So many young men have only heard, outside of a limited amount of church music, popular music, much of which is, politely, insipid. It's like only dining at McDonald's one's whole life. Young men need to have a much better diet than that. We all need to have a good musical diet.
Peace,
Alan