This is a good question and one that shows cultural sensitivity. As you know, Asia is a huge continent. So, generalizations are hard to make. However, here are a few points that will help you get started. I will start with what you should not give and then comment on some gifts that are appropriate.
In China, you do not want to give clocks, as this symbolizes death. In addition, you do not want to give anything sharp like scissor or a knife, as this implies a severance of relationships. These examples are given to show that gift giving is cultural.
Here are some appropriate gifts based on a few occasions. In China, it is appropriate to give a sum on money in a red envelop to children at New Years. Money for other occasions such as weddings are appropriate in China and South Korea.
For other occasions, like house warming, a box of soap is very appropriate in Korean society.
Also here is a word of caution, stay away for anything that have the number 4, as it symbolizes death. Apart from these few rules, generally speaking, all gifts will be appreciated.