It Started With a Kiss is technically a remake of the Itazura Na Kiss (1996), and it's one of my favorite dramas ever.
Leh Ratree is also good. I think the original contained a rape scene, which I couldn't watch, so I like that they remade it and took those parts out.
Hana Kimi (Taiwanese version) is my favorite version of that story, I don't think I even watched the japanese version all the way through, and I did drop the Korean version. There is something about the comedy in the Taiwanese version that I personally prefer, plus the girl actually could pass as a boy. Ella Chen's acting does take some getting used to though.
Kou Kou Kyoushi 2003- I prefer the remake because even though the lead does some not good things (to put it lightly), you can understand his motivation. In the original I just couldn't even like the guy at all, he just seemed like a bad person throughout.
Fated to Love You- I haven't even tried to watch the remakes, I just like original Taiwanese version.
I like remakes as long as they add something new to the story. I hate it when they remake something just because they have a bigger budget or better cameras and they do nothing to change the story. I don't mind a drama that looks "old" as long as the story and the acting are good.