Bridal Mask
Scholar Who Walks the Night
Gu Family Book
I Hear Your Voice
Doctor Stranger
Faith
Descendants Of the Sun
Beautiful Gong Shim - it's currently an airing drama atm. But he protects her and knows martial arts too. It's not an action drama though. It's super comedic.
Bridal Mask +1 (agree with cherrybunny)
The Princess' Man
That Winter, The Wind Blows - it's a melodrama, but one of the better ones I've seen IMO.
I Hear Your Voice +1 (agree with cherrybunny)



Master's Sun (Male protecting the female, rom-com and GREAT chemistry between the leads)
Descendants of The Sun (Song Joong Ki... I don't think I need more explanation ;) haha, I think you will like it if you've seen his acting in Innocent Guy). 
Missing You (in between Healer and Innocent Guy I would say) 
Queen In Hyun's Man (you don't like historic but it time skips to modern and the male goes back and forth. Really interesting, the chemistry is off the chart... They actually dated a while in real life too) 
Please Come Back Mister (fresh story line! You'll end up falling for the 2nd female lead. Really funny drama) 

And if you're willing to watch a foreign drama, I recommend this Chinese drama - Shan Shan comes to eat. 

Enjoy! Let me know what you think! 
Newly released drama "The K2" has been released...you may want to check it out,it stars ji chang wook(healer) and yoona(snsd)...the plot is action thriller and im sure you will like it as much as healer...4ep have been released and im alrdy loving it(highly recommended to watch)
I second I Hear Your Voice and K2.

I Hear Your Voice: Kdrama / Fantasy / Romance / Crime. A MUST WATCH among kdramas. Same main male actor, writer and director as Pinocchio. This is about a boy witnessing the murder of his father. A witness gives a testimony against the culprit and send him to prison. Years later, the boy meets her again. The trick is he can hear the thoughts of people when he meets their eyes. He somehow meets with the culprit's eye and hear that he's seeking revenge against that witness.

K2: Kdrama / Action / Romance / Politics. The drama just started so it's hard to say if it's good or not. But as Loving kdrama said, it's with the same actor as in Healer. This one is about a former soldier hired as a bodyguard for the daughter of a presidential candidate. Lots of action.
Oh Excarius I am in the same position as you!!! Healer is one of my best drama!!It has all I want to see in a drama.

But I haven't find yet a similar drama like this apart from city hunter which is old with a not satisfatory end and kinda doctor stranger. The K2 was similar to healer at the begging but not any more... Oh and I hear your voice has a minus point for me:the romance is not the one like healer's because there is an age gap with the heroin far older than the hero....

Look....If I am right you will like:
a)Pinnochio.It is the only korean drama with a real-down to earth relationship, no stubborn acting because the female lead cannot lie.
b)That winter the wind blows
c)If you want give a try to historical drama Joseon Gunman...

I have check out and see every korean drama that exist (I am a fanatic kdrama) but a one like healer I haven't find yet...

Hope I helped...
Pinocchio. I found it very similar to Healer.
Legend of the Blue Sea
+1 Faith
The Royal Gambler (sorry, those last two are historical)
Sassy Go Go
+1 Beautiful Gong Shim
Another really good mystery drama is The Girl Who Sees Scents. Theres also plenty of action, romance, and comedy. The female lead is protected by the male lead, but she also has a really cool power of her own. Definitely worth the watch!
MDL moderators should make a page for the newcomers on what to do after just watching Healer. I've seen many people ask recommendations for Healer. 

I loved new drama While you were sleeping... kinda similar with the feeling

you could like also Tunnel, not too sweet but a lot of perfect drama, unexpected things

have you seen Goblin? one of the best dorama Ive ever seen

+1 to City hunter

She Was Pretty (hilarious)

While You Were Sleeping

Love 020 - chinese drama and it's Worth watching.

Once Upon A Time In My Heart/ Karn La Krang Neung, Nai Hua Jai ( thai drama 12 eps, a gangster/mafia and ordinary girl, almost similar to Healer)