I did mention Grave of the Fireflies!Me2 with A silent voice, anohana and clannad.
Many mention of "your lie in april" but for me the show had pacing problems and was filled with forced melodrama and overreactions, couldn't relate and overall found the show lacking and highly overrated.
No one has mentioned Grave of the Fireflies yet?, i watched this movie once several years ago and cried like a baby, will never watch that movie again because it's just to sad.
The Troubled Life of Miss Kotoura is actually a fun Comedy with some drama and romance, but the first episode is really heavy on the sad parts, i can't watch this again without breaking up in tears (especially the scene where she just 'breaks' and does a silent scream).
First time watching Aria the animation(3 seasons) i found myself tearing up so many times, not because it was sad but because it was very touching and warmed my heart in the best way
I also tear up in sad parts of supposedly nondrama series. I remember tearing up in a particular scene near the end of the last season
episode 10 or so when Noda was missing Chiaki-senpai
Aria is indeed heartwarming.