Oh don't be spoilsports. Choose something that's really easy to remember/pick up and teach them to say it, then watch and amuse yourself when they repeat it to others.
I teach them "その口じゃない" .

Gingerninja wrote:

The thing that annoys me most is when people ask you to "say something" in Japanese. 

Yes. ok, because you'll understand anything I say anyway.   
Why even waste my time, I'm not a performing monkey.

I used to get annoyed by this too, but after some thought, I realized I may have been the annoying one. Most people probably ask you to say a random phrase because (1) they just want to continue the conversation or show their interest or (2) they want to hear what the language sounds like when you speak it.

(1) is probably the most common reason, and we could still criticize it for being shallow. If someone really wanted to continue a conversation or show their interest, there are much more interesting ways they could do so (but, to be fair, perhaps they simply didn't have time to think of such ways).

(2) makes a lot more sense because perhaps the person has never carefully listened to the language or is curious to hear your voice in the language. People are often surprised when they hear me speak in my Japanese voice ("It's not you!").

In any case, if it bothers you so much, why not follow up by questioning their motives, e.g. ask why they want to hear something they couldn't even understand, etc.?

僕は嘘をつくのは得意ではない、でも嘘を付くこと自体はそれほどきらいではない。変な言い方だけど...