I'm American and I've heard Americans say "I beg your pardon", but like you I've always thought of it as a slightly proper (maybe WASP-y) idiom. People frequently say "excuse me", "sorry?", or "say again?". At least I do. Maybe I should get my ears checked.
They'll also commonly say "pardon me", which is a bit nicer "say again", but definitely nowhere close to "I beg your pardon" uptightness.
I'm familiar with the expression but if an American said that to me, I'd probably think it meant "rethink what you just said".