Which part of Asia? It's rather a large place. And how do you define "Asia"? Some definitions include Afghanistan, Iran, etc. Some don't. Are Japan and India really similar enough to lump together as being in "Asia" in any meaningful way? Do Bangladesh and Korea tourists encounter the same health issues? The question needs more specificity in order to address it carefully.
Common complaints: gastro-enteric, often flu and cold and sore throat. In the towns you normally do not get worse diseases, if you travel to rural or out of the way areas, it might be different.
In India and Sri Lanka I rarely get sick, Bangla Desh is dangerous and Myanmar is sometimes also problematic.