Heh, Oldboy.
It will not make you cry. But it's the most angsty and tragic one I've seen.
And you're specifically asking for a k-drama or k-movie, but if you are in the mood to cry, try the anime movie "Hotaru no Haka" aka "Grave of the Fireflies". It's Japanese, and it's animated (please watch specifically the animated one, not the live version one), but I can guarantee you'll cry BUCKETS of tears.