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Tear Jerker: K
For the entirety of Episode 6.
Tatara's death, for one.
Shiro going back home only to find that it doesn't exist.
Shiro asking Kuro to kill him once he realizes that all of his memories of being an Ordinary High-School Student is nothing but a lie, which means he could really be a murderer without knowing it.
In episode 10, the flashback with Yata and Fushimi. You could feel how desperate Fushimi is to get Yata's attention and even burns the tattoo in front of Yata to get Yata to hate him. This analysis right here might hit home...
The ending of Fushimi and Yata's Drama CD. It really shows how much of a Jerkass Woobie Fushimi is. It especially hurts when Yata gives him a mild "The Reason You Suck" Speech telling him that Fushimi should just move on in life despite having nobody who cares about him. And it can be so clearly seen how they both are hurting from what the other had done, but neither of them could understand each other well enough to patch it up.
Most scenes with Adolf K. Weismann, from how he began his research into the Dresden Slate with the notion that it'll bring happiness to everyone - only to be selected as the First King with the power of immortality, cursed to live as his sister dies -, to how he voluntarily exiled himself from the world for over half a century (probably longer if not for the Colorless King stealing his body) just because he'd rather not involve himself with anyone that he'll eventually have to watch die. His relationship with Kokujouji Daikaku is also particularly tragic given their recent conversation in episode 12 - his oldest friend waited for him for over half a century, only to hear the words "I'll have to say goodbye again..."
Just about all of Episode 13.
Mikoto's death. The thought that he could have renounced his throne to avoid it but chose not to for the sake of vengeance (which he ultimately succeeds) makes it the more heart-wrenching. Anna's reaction and HOMRA's "No blood! No bone! No ash!" repeatedly, while crying, were particularly heart-wrecnching. Even Reisi Munakata looks obviously pained after killing his old friend despite his attempt to keep stoic in front of his men.
Shiro/Weismann's death. He has finally found a reason to live on the ground alongside humans but ultimately choose to fulfill his responsibility as a king for the first and the last time to save their lives. Both Kuro and Neko's reactions are tear-jerking and Kukuri, even though she doesn't remember, can clearly feel someone is missing.
Most of the lines shown by the second season preview sound depressing.
The final chapter of K: Memory of Red. HOMRA is planning to surprise Anna on her birthday which is on 8th December, then you realize that Totsuka was fatally shot on 7th December at 11:45pm. The chapter goes on to show how everything led to the murder, expanded on Totsuka's death scene and the aftermath which leads directly into the opening of the anime. And you will know by now that Anna will soon proceed to lose Mikoto shortly after losing Totsuka...
Every flashback of Misaki and Saruhiko is just incredibly sad. When you bring them into chronological order (including the ones from CD Dramas and the Mangas) you can watch their formerly wonderful relationship crumble and then brake into pieces without Misaki noticing and Saruhiko unable to stop it.
Watching them interact when you know that they used to think that what belonged to one belonged to the other as well is heart wrenching! The fact that they obviously suffer and miss one another doesn't make it any better.