I've seen Great Teacher Onizuka recommended here. WTF???? Seriously: WTF?
Anyway, as for me, well, I haven't cried watching anime, no. But a couple of series that impressed me (perhaps because they were some of my first) were Elfen Lied which you already saw, and
Kannazuki no Miko which is quite hated around here, but I found it pretty good, if you can get past some bad episodes on the beginning and a couple of stupid things.
School Days is not tragedy, it is comedy.
Kanon is one thing you should surely watch. That was truly tragic, and a damn fine series.
Kimi ga Nozomu Eien is pretty good and surely is tragic, although I do not understand at all how people has found it so unbearable and devastating... It left me quite cold (I saw it a couple of weeks ago). Perhaps that's because as time passes by, less things I watch are bound to impress me.
Another classical recommendation is
Iriya no Sora, UFO no Natsu. Again, pretty good, pretty good, but didn't kill me.