"Far East food doesn't agree with me" would not commonly be used in everyday English conversation. I would understand what you are trying to say (since in Spanish we say something similar), but it would still be odd to hear. It would be like saying that you and Far East food don't agree on something, such as a certain topic. It is like saying that you and Far East food argued and that it did not agree with you. People would (or might) still understand, but again, it is not common at all, and it's meaning doesn't really make sense (even though people may understand what you are trying to say). So this sentence would in the end be considered "Incorrect", or just not a really good use of words.
Saying "Far East food is not good for me" would be considered correct, but it depends on what you are trying to say. Saying this means that Far East food is unhealthy for you. If you said this, it would mean that the food is either unhealthy for humans to eat in general, or that you are allergic or have a bad physical reaction to that type of food. Don;t say this if you are just trying to say that you don't like it because of personal preference. This would be said if you didn't like Far East food, because of some past experience where it made you feel sick or made your stomach hurt.
If you just want to say that you don't like a certain type of food because it makes your stomach hurt afterwards, or because it makes you fell sick, you can also say "Far East food doesn't go down well in my stomach.". This is commonly used and people would know what you mean.
If you want to say that you don't like a certain food just because of personal preference, you simply say, "I don't like Far East food."
:) Hope this helped!