There's been a couple of songs that made me feel SAD and.or emotional, if not made me cry.
A lot of songs from the Witch Hunter Robin soundtrack (my FAVORITE anime) are sad, especially "half pain" by Bana (the ending song). Then there was "Fukai Mori" by Do As Infinity (InuYasha 2nd end song), "Every Heart" by BoA (InuYasha 4th end song), and (although it's just more emotional than anything) "Hemisphere" by Maaya Sakamoto (the RahXephon opening song). But Witch Hunter Robin has some of the saddest songs I've ever heard.