I don't really cry a lot when watching dramas but the once that made me cry or really emotional are...
49Days (Korea)
Boku no Ita Jikan (Japan)
Gloomy Salad Days (Taiwan)
I am Sorry, I Love you (Taiwan)
Rooftop prince (Korea)
Moon that embraces the sun (Korea)
Ouroboros (Japan)
i miss you
cruel city
Reply 1988, bc im kinda weak for family drama, so this drama really touch my heart for their family bonding 
Meteor Garden
Meteor Garden 2
The Hospital
Fated to Love You
Autumn's Concerto
In Time With You
The Way We Were (faucets of tear, I swear!!)

^they're all Taiwanese since I mostly watch TWdramas :)
Weirdly enough it was Reply 1997 and Secret though my tears were mixed with a lot of anger...
as for me, the drama/movie that i remember i cried when watched it:
-one litre of tears, MOTHER, koizora, Boku no Hatsukoi wo Kimi ni Sasagu  (Japan)
-stairway to heaven, autumn in my hearts, Nine:Nine Time Time Travel, Iljimae  (Korean)
-battle of changsha (China)
There are a lot, but the one I cried the most is definitely Tree of Heaven T.T I can cry even now when I think about it, and it's been a couple of years xD
Haven't watched too many, but I cried for like three days after Rooftop Prince and My Lucky Star ended.
I'm currently watching "Remember - Son's War" and that drama is making me cry a lot!!
I don't really cry for dramas, tbh. 
The only ones that ever really got me going were Secret Garden (pink sweater. you know the scene I mean. TEARS EVERYWHERE. IT WAS GLORIOUS) and 49 Days (the ending of it just got to me. Loved it though.)
'Mop Girl' had me crying a bit too, but I think I might have just been emotional when I watched it. 
I don't really cry for dramas, and now it's making me a bit upset that I can count on one hand the amount of times they've made me cry D:

I'm much more likely to cry for movies- 'A Moment to Remember' left me an emotional wreck, and I'll always treasure it for that. 'Classic' was pretty bad too, but it wasn't an-entire-box-of-tissues bad. 'Harmony' was though. 
Scent of a woman!!! Must've gone through 1 whole box of tissue!!
I would recommend kdrama "IM SORRY I LOVE YOU" staring by so jisub and im soo jung. I have watched a ton of korean,japanese and taiwan drama. But this is the only drama that made me cry like a crazy person. After finished watch drama, im still want to cry just by listen to the ost. Just prepare tissue when watch this drama especially during ep 9 -16 . The male lead and female lead really doing a great job!!!!
Tonbi. I couldn't stop crying in every episode. The father and son relationship is so heartwarming