Cried? Never. Quite a few tugged at my emotions though. Off the top of my head:
Cowboy Bebop
Katanagatari
Code Geass
Death Note
Gungrave
Japan in general and animes specifically have a penchant for the dramatic. However, those 5 (that I remember off the top of my head) did it very right. Hit you deep, you know?
Then there are animes that hit you with the strong stuff midway through and leave you to recover, like (MASSIVE spoilers bellow):
[spoilers=Death Note]L's death.[/spoilers]
or
[spoilers=Macross Frontier]When Michel is killed while Clan is in the tank and forced helpelessly to watch him die to defend her.[/spoilers]
[spoilers=Code Geass]Code Geass is just one emotional punch after the other... Hard to name ONE moment, the whole thing is an emotional rollercoaster.[/spoilers]
Also: obligatory tvtropes link [http://tvtropes.org/pmwiki/pmwiki.php/Main/TearJerker].