49 Days and Cruel City although I cried really nasty in Moon Embracing the Sun and Dream High too. Well I guess I am a sucker for the cross dress ones too like Coffee Prince and You're Beautiful. I think the first ones to make me tear up a lot were Goong and Boys Over Flowers. 
As for saddest Korean movie, it would have to be Unforgettable; and for drama, it would be Uncontrollably Fond. But killed me in the end, but it's ok I didn't want to use my heart anyway...
I saw A Thousand Days Promise recommended........you leave that show in a state of depression.  Soo Ae destroyed me with her acting.
I cried for 2 whole episodes of Fated To Love You.  I know it's not a sad drama really, but in the middle bit when they go through trauma (like in most dramas) I knew it was coming, and I cared so much about both of them.  I didn't want them to hurt, and I just cried the whole way through the traumatic stuff.  Such good characters.
Marriage contract and I'm sorry I love you
Kill me Heal me, it was really the best kdrama and so sad and interesting with a fresh plot
Chuno- damn I woke like a panda..
Cruel City-Life is Cruel to our Doctors Son
Fated to Love you(Korean)- I cried to all the sad moments
Memories of Bali was memorable and sad for me...I will watch 49 Days soon so i can relate
Gonna go out on a limb here and say Mimi
I have just watched a few kdramas but "Kill me Heal me" was just so good, I cried so much
Scarlet Heart Ryeo
Missing you
Fated to Love you