As you can tell from my statistics I watch A LOT of dramas. I got into Asian dramas almost by accident. I have always enjoyed anime and had started watching it again in 2007. In 2008 I discovered that some of the anime I had watched also had live action Japanese dramas. I watched a few of them but I wouldn't say I was hooked yet. Then I came across Hana Yori Dango and really liked it. So much so that I went and watched the Taiwanese version also and at that time Boys Over Flowers was starting so I watched it too. While watching it I saw that another Korean drama, "You Are My Destiny" was near the top of all lists so I started watching it which introduced me to Yoona and Girls Generation and KPOP. After that I was hooked and have watched over 300 so far and counting.
The thing I like about the Asian dramas over our (USA) shows is they have, in most cases, one season. A clear beginning and ending. Your not waiting season after season after season and potentially missing out on other good shows. I also think that a lot of Asian actors tend to be more seasoned because they are able to do more than just one role year after year. It's like the golden days of Hollywood when when an actor would turn out 3 or 4 movies in a year, or more.
Last thing I'll mention is my list. If you look through my list you'll noticed I don't give out a lot of 10/10. A drama really has to not only be unique, but for me it needs to make me want to watch it over again in order to get a perfect score out of me.