If you're born a dual Citizen then it's not a problem. If on the other hand a non-US citizen applies to become a US-citizen then they will be asked to "renounce all other allegiances". However depending on what your other "allegiances" are the US will not bother too much about following up on it.

Also if you volunteer (as opposed to being conscripted) for military duty in your non-US nations then the US can kick up a fuss.

But basically if you're a dual citizen with a nation friendly to the US then you really have nothing to worry about.