I am an Orthodox Christian, and everything below should be understood in this light. So when I hear religious music, I think of Orthodox religious music, particularly Byzantine chant.
I believe that as far superior classical is to pop, so Byzantine chant is to classical. Those who know my opinion of pop will understand how strong a statement this is.
Of all the classical composers, the one who affects me most strongly is Haydn. His is the vision of the sublime, whose music I find astounding. However, even his music still sounds frustratingly earthbound when I compare it with Byzantine music well-sung (I hear it sung badly often enough). So does Josquin, Victoria and other western musicians I admire. To have heard Byzantine music in the full glory of Constantinople would be a dear wish of mine, if it were possible.
It seems to me that the greatest perfection must be in a single line: even a fifth or an octave are merely perfect consonances, and as for a third... For if we believe perfection to exist, why introduce imperfection into our vision of perfection? Byzantine music is at its heart monodic (single line). Although now there is an 'eson' (drone 'bass' type of thing), this is thought to have been added after the Ottoman invasion of 1453. To have it now is possibly a concession to our spiritually weaker times (so I would say).
This is the Easter troparion (song), possibly the best known tune among Greek Orthodox Christians. It is sung from Easter to Pentecost frequently, and everyone looks forward to singing it on the Easter vigil.
This is also very peaceful, if quite long.
Polyphony and in general non-monodic music was rejected in our Church by various councils, in Greece at least. However, Russian liturgical music has more than one part. Recently Serbia changed from its Russian practice back to a Byzantine practice. The change in Russian music was criticized both in the Greek Church and saints within the Russian Church. These latter pointed to the older Russian chant as more beautiful, and more spiritual. However, that is not to say that Rachmaninov's (for example) church music is not very beautiful, it is, or 'valid' religiously, having been approved by various Russian synods. It seems to me that God has given us such music because in our weak times we need something slightly less ascetically spiritual, in order for us to understand it, and benefit from it to perhaps a greater degree.
Rachmaninov's music has already been posted above
Another type is Romanian Orthodox music. All Romanian singers I have met happen to be very good, but this could be coincidence. This seems to me to bridge the gap between Byzantine music and the modern day very well.
It has more harmony than traditional Byzantine music, but still is really based on a single melodic line.
At the heart of Orthodox liturgical music are the words. These must be easily understandable: virtuosity is not desirable for us. I have not studied this; these are my general impressions really. I hope to have the opportunity to do so and learn more over the next few years.