Tear up? Well we'll go with that feeling at least, if not the act.
http://www.youtube.com/watch?v=VmSjrr0XQNw, less for the intrinsic song itself and more for the memories it invokes of that horrible, horrible, soul rending experience that was Clannad.
http://www.youtube.com/watch?v=Dtx3vQI1Rk4&feature=related, from Shawshank Redemption.
http://www.youtube.com/watch?v=03qBqP2I4p8, http://www.youtube.com/watch?v=kKAeG8fy1fg, and http://www.youtube.com/watch?v=0V-WGwJJj58 from Cowboy Bebop. All are quite powerful pieces I think.
http://www.youtube.com/watch?v=SmVAWKfJ4Go, by Johnny Cash.
http://www.youtube.com/watch?v=Yp3sxC7oVhk, from the Halo 2 soundtrack. One of my favorite "put on repeat forever" melancholy songs.
http://www.youtube.com/watch?v=sSH8b33wZcw, from The Lord of the Rings musical. Don't knock it, it's good music.

http://www.youtube.com/watch?v=z_73Cc2typk, by Stan Rogers; and http://www.youtube.com/watch?v=OiBnrGHeBOY, by Loudon Wainwright. Both are better when you pay attention to the lyrics.
http://www.youtube.com/watch?v=3jP4Osvgvos, from the Ice Age soundtrack. Another "put on repeat forever" song I like.
http://soundcloud.com/piercingsight/solitary-sonata-tiara#play and http://soundcloud.com/piercingsight/stairway-to-madness-diane, by Steven Mortensen. Hell most of his http://soundcloud.com/piercingsight/dont-look-discord.
Edit: Oh and http://www.youtube.com/watch?v=sgxLsb-Lztc, by Hevein.
Edit2: One more thing! http://www.youtube.com/watch?v=G-Vg2YS-sFE, by The National.