Of all the Asian bicycle touring, japan and China are my favorite. Bicycle culture is strong in both countries and bicycling mix with traffic is just better. Japan is clearly the most civil, but China still has bicycle lanes that connect major cities as well as for inner-city traffic.
Frankly, the cities in Korea and most of the areas around the cities make no accommodations for bicycles. Bicycling in any city in Korea is suicide. In addition to the roads being designed for "automobile only", a lot of Korean drivers are bullies - feeling that the prestige of the automobile gives them privelage and clout over bicyclists and pedestrians. It is common to have Korean drivers pull up right behind a bicycle and honk and intimidate you until you jump off the road. They will also physically and intentionally run you right off the road. IMO, Korea ranks very low on the places to bicycle tour.
For industrialized Asia, Japan is better than Korea and probably better than Taiwan. There are many alternative routes, the drivers are more polite and conscious of bicyclists, and frankly, Japan just has a lot more cool stuff to see and do than Korea.