There's a fine line between "tearjerking" and plain ol' "depressing" as in down-in=the-dumps anime. Not once did I ever felt depressed watching Clannad as it was born out of empathy for the characters. School Days? I mean seriously, who didn't foresee Makoto's death coming from a mile away? And Naruto isn't depressing - it's just lame. Substituting shitty anime for "depressing" is not what the TC had in mind. Sure, the fact that the anime didn't deliver was depressing, but that doesn't mean the anime itself is depressing.
Some shows off the top of my head:
Wolf's Rain: this is seriously depressed me nearing the end of the show. Characters that I thought would live died, even all the human, minor characters. Gah, what a depressing anime. Still, it was a very moving anime about wolves searching for a paradise that would accept them. Like Pandora's box, there seemed to be hope at the very end of the last episode.
Victory Gundam: By far, this show has got to have the highest amount of onscreen deaths of minor characters featured in the entirety of the Gundam franchise. I mean, the entire Shrike team, whom we get some time to empathize with, gets completely eliminated. God, this was one depressing show that I would never, ever want to watch again.
Casshern Sins: This show reeked of depression from the getgo. Since the show is all about death and "rust", many scenes of decay, deterioration, and death were featured as well as the breakdown of all civilization into chaos. Even Luna, the supposed symbol of life, turned into an icon of despair and calamity where she bestows death on all those who offends her. A slow series, but very touching. The animation style helps bring out scenes that are pretty damn depressing.