japanese, korean, taiwanese, chinese as long as its asian...

These are the only ones that come to mind:
Celeb to binbo Taro - a multimillionaire girl falls in love with a poor man with three kids
Tatta hitotsu no koi - a well brought up girl falls in love with a poor gu
Hi, my sweetheart - a rich girl falls in love with a silly guy who is suppose to be very poor. What she doesn't know is that he is in fact the only heir to a large fortune
I second Tatta Hitotsu no koi

You Who Came From The Stars: well obviously he's not exactly ordinary since he's an alien, but to everybody else he's a normal guy, and she's a superstar

Queen In Hyun's Man: same case as the previous, only he's not an alien but a guy from the Joseon era :p

Que Sera, Sera: the rich girl does fall for the ordinary guy but... it's kind of twisted. I can''t tell more because I don't want to spoil the drama, but let's say the drama isn't a pleasant romantic story about how the ordinary guy gets the amazing girl...

My Sassy Girl (movie): she's not rich, but she's rather extraordinary, while he's painfully normal

I think it may also be the case with dramas like Witch's Romance and especially Secret Love Affair, although I can'tbe sure since I haven't watched them
The first two that came to my mind (both Korean) were The Snow Queen and A Love to Kill. However it is worth mentioning that these are both melodramas and not everyone's cup of tea.
That Fool (Korean)- She's a top-star while he's a low level post office worker.
The Birth of the Rich (Korean)- She's a wealthy heiress and he is a hotel bellboy who believes he's a chaebol's son.
History of A Salaryman (Korean)- She's a spoiled heiress and he's unemployed hoping to become a salary man.
Ugly Alert (Korean)- She's a chaebol and an executive in the company while he's an ex-convict who took the blame for a crime he didn't commit.
Take Care of The Young Lady (Korean)- She's an heiress and he's a con-artist.
101st Proposal (Korean)- She's a successful announcer (not exactly rich) and he's a near-unemployed , highly skilled, craftsman.
I Love Lee Tae-Ri (Korean)- She's a chaebol heiress while he's a 14 yearl old boy who gets transformed into a 25 year old man.
Hayate The Combat Butler (Taiwanese)- She's a beyond wealthy heiress and he works odd-jobs to pay his parents debts.
OMG, Yes!!!! Tatta hitotsu no koi is a third for me! I agree. This one is good! :D