Hello, I've always been a sucker for romance-related Asian drama and I particularly like it when the plot features more than one couple, preferably where the other couple/s are also important and aren't just for comedy relief but get their own story. If you could suggest me a Chinese, Japanese or Korean drama than I'll be very thankful. The main genre can be any; historical, comedy, school etc. but has to have the romance part in it.
Here are some I have watched for those of you who also have a similar taste to mine.
Love Shuffle (JDrama)
Hero Dog 2 (CDrama)
Boys Over Flowers (KDrama)
My Sunshine (CDrama)
Thanks again in advance :)
Heartstrings (K-drama)
City Hunter (K-drama)
Cheese in the trap
That Winter, The Wind Blows (watched first 7 episodes so im not sure how it unfolds but there's definitely something going on)
Reply 1997
Switch Girl
Suki na Hito ga Iru Koto
Marriage, Not Dating
Mask
Scholar Who Walks The Night
Secret (not ship-worthy tho)
The Princess' Man
Gu Family Book
Love That Makes You Cry

from what I've heard but didn't watch also;
Six Flying Dragons
Descendants of the Sun


Reply 1988
Reply 1994
Protect the Boss
Coffee prince
Faith
Love Rain
Secret Garden
Six Flying Dragons
Warrior Baek Dong Soo
You're all Surrounded
The Lover
Ugly Alert
Descendants of The Sun. Second couple has their own issues/storyline and ok screen time.
Secret Garden.
Coffee Prince.

49 Days, but this one is a bit different. Not sure if it'd be up your alley. The relationships are kinda intertwined and it's not exactly a happy drama.
Descendants of the sun
High Society
One more happy ending
Pride
Can we get married
Oh my venus
If you liked My Sunshine, maybe Boss and Me (Shan Shan comes to eat), the novels were written by the same author
Descendants of the Sun   - two OTP couples, no annoying love triangles
High Society
A Gentleman's Dignity   - have more than two couples, very funny
Bluebird House   - two couples, but its a long drama
D-Day
Doctors
Reply series - 1997, 1994, 1988 (the last was my favourite)
The Producers


You already got a lot of suggestions here, but I just wanted to add the j-drama Slow Dance. It's a good one, with likeable characters, and for some time I really didn't know who was going to end up with whom. :)
All great suggestions - would recommend something older but one of my favourite J-drama rom-coms - Love Revolution (about 3 female friends and how they find love). You can also try Better Man - a recently completed Taiwan drama about how three brothers (supposedly very eligible bachelors) and how their love lives transpire.   
I just recently finished Love in Time, and it has a side romance (between a vampire and a fox). :)