I cried a lot while watching Missing you and Fated to love you (Korean version). Rooftop prince started out hilariously, but then the plot thickened and it was a real tear jerker.
The 1st thing that came to my mind was Cruel City T______T
The first drama that made me cry was Reply 1988 and the second was Signal.  but only a little bit.
I remember crying alot when I was watching 49 days and Akai ito.
I think the 2 Korean dramas i found made me the saddest were, 49 days and Empress Ki. I had such an emotional attachment with these 2 dramas and I cried so much watching them.

I can't even name the amount of Japanese dramas that had crying my heart out.... there are way too many!
Marriage Contract had me crying all the time
49 days!  Loved the drama!!
Innocent man and Ugly heart (although this is a really long one) made me cry.

I see a lot of people recommend 1 litre of tears. I tried watching that but Japanese dramas just in general seem cheesy to me. I don't know exactly why but I just can't watch them. 
Nine Times Time Travel will make you cry, plus it's awesome. 
49 days is basically a drama about crying, so it ought to be recommended, but I don't remember liking it too much...
Bridal Mask is also really sad. 

But honestly, when it comes to crying, I go in Jdramaland. 
Autumn Tale, Stairway to Heaven, My Rosy Life, and Jewel in the Palace

*There are plenty in Japanese dramas/movies thou
Autumn Tale
Hong Gil Dong
Flowers for My Life
Incarnation of Money

The first 3 can hold their own against any sad J-drama. Incarnation had sad moments.
 
49 Days... I cried like a little baby...
Empress Ki
The Bridal Mask
49 Days
Secret
A Thousand Days Promise