Sephardic music (music of the Spanish Jews which was spread all over the place when the Jews got the boot) is really worth a listen.
Rondinella - Sephardic
Hadass Pal-Yarden - Ladino
Islamic Music:
Sabri Brothers - Qawwali (Sufi) music from Pakistan
Traditional Whirling Dervish - Sufi music from Turkey
Mercan Dede - modern Sufi music (Turkey and US, with some english lyrics)
Buddhist Music:
Monks of Drepung Loseling - Tibetan Buddhist
Gyuto Monks - Tibetan Buddhist
Vocal music is usually best when the words are in a language I don't understand.
I was going to mention Qawwali. There's also the kind of stuff Dead Can Dance or Azam Ali does, which I'm pretty sure has religious connotations. Loads of "world music" is religious, but if you don't understand the words it doesn't really matter!