Many an anime series has made me cry, as have some games (and considering what forum we're on, I was very surprised at the lack of a game option). The Japanese company Key Software, to take the most famous example, is a game studio that could easily fill a moat or three around their offices with all the tears shed by their products' consumers. Yes, they make those dodgy games whose main objective is to get the male player character into the heart/pants (they used to make the latter category, then went all-ages and released extra "no sex please, we're Japanese" versions of their older products) of one or more female NPCs. And yes, their art style may take getting used to - many characters look very underage, and occasionally their upper lip seems to be horizontally aligned with their lower eyelids. But their storytelling is utterly brilliant, as is their soundtrack department. Three of the four most successful games they've released so far (Air, Kanon, Clannad) have spawned anime series, all of them have stimulated my lachrymal glands to release watery fluid, and an adaptation of the fourth one (Little Busters!) seems not too far off.
Apart from game-based series, I am also not ashamed to admit that I shed a few tears at a Sailor Moon season finale (though I forget which one - it was just too long ago), and also at several episodes of Fruits Basket even on the fourth re-watch.
But not only animated Japanese stories make me cry, oh no. There's also the wonderful work of manga to squeeze the tears from my face. I had great difficulty reading the final chapters of Love Hina because the watery film constantly covering my corneas made the speech bubbles very blurry.
Leaving the wonderful world of Japanese fanboy culture, I have also shed tears at episodes of ER (does the name Lucy ring a bell for anyone?) and even Futurama. The bastards, the complete and utter bastards. Ahem.