It's probably telling that the first few pieces that came to mind for me are instrumental.

As far as songs go, if they move me for themselves, it's generally by the combination of the music, the lyrics, and the way they're performed. Lilith Fair's Hot White Cum will move me to a kind of bouncy, aggressive cheer that's as much a function the major key and upbeat rhythm as the humour in the childlike frankness of the words. Amanda Marshall's Love is my Witness has this chorus that I find spiritually moving, even though the song itself is a romantic one, because Marshall's voice soaring into the upper registers flips some internal switch in me to interpret the chorus in a personal-responsibility-meets-global-consciousness kind of way.

It'd be impossible for me to list even a fraction of the songs that do something like this for me. And then there are even more songs that move me not for themselves but because I associate them with some memory that has an emotional component of its own, which may be completely unrelated to the actual content of the song.

The mind is a fascinating place.