Hmm...here's some that does it for me.
I know you by heart, Waly Waly (except for that screeching bit in the later part of the song - else it's gorgeous through and through!) - Eva Cassidy
The Last Rose of Summer, Shima Uta (the duet version with THE BOOM) - Izzy
Song of the Sandman - Enya
風のように - S.E.N.S (not sure about the English title - perhaps it is the same piece mentioned by Braxton??)
These songs don't exactly make me weep but they are moving in a very fundamental way.