It seems to me that people posting here put too much emphasis on using idioms as a shortcut to sound "like a native speaker." I think this is misplaced. As I learn Spanish, I am becoming intensely aware that, in between single words and idioms is a vast ocean of short phrases that ought to be mastered long before trying to use idioms. There are so many situations in which a native speaker says "That's perfectly correct, but it's not the way we usually say it."
You want to be particularly careful in a culturally sensitive area like talking about the body, feeling sick, and so forth.
In the US, "something I ate didn't agree with me" is really very good. It is nicely in between formal and informal, specific enough but not "too much information." It says enough, but not too much. I think it works in the UK as well as the US.
Some other straightforward things I might say, in real life, when speaking to someone who is neither a stranger nor a friend.
a) "I think I'll skip lunch today, my stomach doesn't feel right."
b) "I wonder if there was something wrong with the chicken. I feel a little queasy."
c) "I have a stomachache, nothing serious."