Originally Posted by HkCaGu

Nobody can provide an answer without knowing which country it is. It is that country's policy that matters, not the US'.

Assuming that you...

...want/have to use your home country documents (which is apparently not your case): You may have a conflict if your home country enforces the same elements as the US, such as advanced information. For partially different names, airlines can accept multiple identities at check-in but not booking. For totally different names, you simply cannot board one round-trip itinerary at all. And if your home country prohibits dual nationality, you may need to hide your US citizenship by using two separate itineraries with a stop in a third country.

...want to hide your identity from your home country: The US passport must contain your place of birth and date of birth. (The Japanese passport, e.g. does not contain POB.) These may be too much already (if not also the technology mentioned by other posters) to get you questioned.