No music has ever made me cry of itself, probably because of my unsatisfactory upbringing, but certain things bring a lump to my throat and I can get myself to be tearful if I think of certain things while listening to them. A prime example would be Litanei auf das Fest aller Seelen* by Schubert, especially the version by Janet Baker. Mon Coeur S'Ouvre A Ta Voix (Saint-Saens) sung by Maria Callas (practically anything sung by Callas is devastating to me), whilst in another genre, Go Leave by Kate McGarrigle and Flowers and Wine by Pete Atkin (lyrics by Clive James) are very moving, if the last is a bit studenty.
*Try typing this into youtube to see a very moving montage about The Troubles.