Toboe accidentally killing his grandmother. Watching the changes in his body language as it slowly dawns on him that she's dead is heartrending.
The old wolves being essentially enslaved and worked to death in Episode 5. Pair that with somber, melancholic cues from Kanno's score and you're going to need a lot of Kleenex.
There is also another one earlier on the series when Jagara kills Harmona and Darcia goes totally insane, first dancing one last time with Harmona's corpse, then completely going mental and smashing his forehead against a pillar several times. A remarkably powerful (and unexpected) scene.
Hige's anguish over even the possibility that he had betrayed his friends. Seeing him go from cheerful to completely wracked with guilt for the last few episodes of the series is really hard to watch. Made even worse by the fact that it really wasn't his fault and no one blames him anyway. Tsume's the only one with any hard feelings, and even that's implied to have more to do with his past then anything Hige might've done.
The death of Toboe. This is only one of several tragic deaths that occur during the show's 4-episode OVA conclusion, but is arguably played (or possibly just a little overplayed) by the writer and director for maximum emotional value. Toboe dies unsuccessfully trying to defend Quent, a man who has always hated wolves. Only at the very end are the wolf and human reconciled. As he dies Toboe dreams of being reunited with the old human "Granny" whom he loved but accidentally killed. Later on, Jerk with a Heart of Gold Tsume loses it and weeps for his fallen friend, which is the point at which if you haven't shed a tear already you will. And then (after a bunch of other stuff happens) that closing theme finally kicks in... "Been a long road to follow..."
"Yeah, I know, kid. I know."
Topping that scene off is the soprano renedition of Toboe's theme, followed by Cheza's lullaby.
Blue begging Hige not to leave her because she is so terrified of Dying Alone. The expression on Hige's face when they embrace is even more heartbreaking, because it looks like he knows they're probably not going to come out alive. But he keeps his promise.
That doesn't even compare to what happens in the series finale. If Tsume having to put down Hige didn't make you cry, Cheza disintegrating to seeds in front of Kiba surely will.
The fact that Kiba - the character who will fight to the very end, who won't let injury or adversity get in his way, who will go to the ends of the Earth to find Cheza - has to watch her disappear just as she was in his grasp and, for the first time, lies down and gives up completely, was just painful. Then the final blow comes when he forces himself to keep walking, only to fall through the ice and drown.
This music video. Seeing all of these couples and their development throughout the series and remembering seeing all of their deaths makes it almost impossible not to cry.
This one too, but for different reasons. It's basically Kiba having flashbacks of his life as he slowly bleeds to death.