I've watched many light-hearted dramas recently and now I'm in the mood for some heavy stuff. It could be either a crime/mystery drama or a tragic/psychological one.

Some examples: Signal, Tunnel, Gap Dong, Boku no Ita Jikan, etc.

Remember: War of the Son
Cruel City
Duel
Secret (2013)
Lookout/The Guardians
Liar Game (both Japanese and Korean versions are great)
City Hunter
Criminal Minds (Korean version)
The K2
"Save Me" is currently airing and it's crazy. It's frustrating and makes you want to do all kinds of things to the cult. Agree with the dramas above as well. I especially enjoyed "The Guardians" and am planning to watch "Cruel City" as soon as possible. :)
If you want something similar to Boku no Ita Jikan you should watch 1 Litre no Namida (2005). It's probably one of the most popular tragedy dramas, and for good reason. 
Voice
White Christmas 
The Guardians 
Circle
Remember : War of the Son
1 Litre of Tears 
Uncontrollably Fond
cruel city 
missing noir
queen of ambition (very recommended)
the winter the wind blows
it's okay that's love
secret 
beautiful life (Japanese)
49 days
love me if you dare (Chinese)
lee jun kai (Chinese)
five fingers
golden rainbow
the devil
shark
last friends (Japanese)
innocent man
secret 2013
Second Love Me if You Dare. It's a really great one, too.

You're All Surrounded was way heavier than I anticipated. It's also a crime/psychological drama.
If you are interested in a heavy movie then I would recommend Always. It's a type of a production that makes you feel all sorts of things.
adding  the dramas:
Autumn Tale
When a Man Loves
Defendant
Padam Padam
I'm Sorry I Love You
What Happened in Bali
Missing You 2012
Scent of a Woman

Movies:
The King
Late Autumn
A Man and A Woman
Kang Koo's Story
The Treacherous
A Moment to Remember
Thanks for the recommendations!! :) I've already seen some of them, but I will definitely check the rest of them out!

(I didn't expect so many, so I guess I will bookmark the page and come back to it whenever I'm in the mood for heavy dramas)
1 litre of tears
Voice
Contract Marriage
Maou
Kazoku game
Liar Game
Crisis: Kouan Kidou Sousatai Tokusou-han
Ryusei no Kizuna
Lookout
Koizora
I'm sorry I love you
Nagareboshi
Innocent man
N no tame ni
White Christmas
Secret
Cruel City
LIFE
Save Me
Ningen Shikkaku
Last Friends
Zeni Geba
49 Days
Boku no Yabai Tsuma
Lipstick
God's Gift - 14 Days
Kamisama Mou Sukoshi Dake
Liar Game
N no Tame ni
The Quiz Show 1/2
The Village: Achiara's Secret
Yume no California

Movies:
Bleak Night
Pluto
Helter Skelter
@iwyn  I am bookmarking this too!   :)
guys i'm bookmarking this as well .. i'm really proud that most of the comment-ers are my chingus