Ojamajo Doremi - it's a slow burner but some of the episodes will hit you like a truck. Cried like a little bitch after the 12th episode of Naisho and at the ending of Dokkaan.
Ie Naki Ko and Les Miserables - God hates children the anime. Both are historical anime about the trials and tribulations of poor children.
Honey and Clover - especially has an impact if you're around 20 and have just finished college. Solanin is a good manga equivalent for this.
Casshern Sins - cyborg goes around watching other cyborgs decomposing in a dying world.
Seconding Mushishi. Natsume Yuujinchou and Hanada Shounen-shi are similar, so give them a try too.
Bokurano - death and suffering the game. Also giant robots.
Stuff like Texnolyze and Jin-Roh might work too, but they have more this nihilistic emptiness thing going. If that makes you cry, go for it. Serial Experiment Lain and Haibane Renmei may have a similar effect.
WWII-themed movies like Hotaru no Haka, Glass no Usagi and Ushiro no Shoumen Daare may do the trick if you're into that.
Jigoku Shoujo, Shigofumi, Death Parade are also somewhat depressing, with an ironic twist on morals. Humans are scum kind of thing. The movie Colorful utilizes a similar theme.
I can vouch for some things from the chart to be genuinely depressing, but some aren't exactly great (A Kite, Hantsuki, Iriya, True Tears), so proceed at your own risk.
