I don't think I ever full-on cried watching anime, but I do tear up a little, here and there. The first time was when Ash let Butterfree go in Pokemon. Some other times include Goku sacrificing himself during the Cell Game (I was a lot younger then), Ryoko seemingly dying in Tenchi Universe, Sylvie dying in Bubblegum Crisis, the end of Grave of the Fireflies, a few times during Cowboy Bebop, Lust's death in Fullmetal Alchemist, the final two episodes of Now and Then, Here and There, and several times during LotGH. Probably some other times that I'm forgetting, as well.
So I guess anime has turned me into a total pussy.