I'm a dual citizen of US and Russia. Being a dual citizen I have two passport both of which have different last names.
I bought a round-trip ticket with the name on my US passport. Will I have a problem getting on the plane?
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Call your airline, but you should not have any problems boarding the plane.
If you have US citizenship, you have to have your US passport when leaving or returning to the US. (http://www.law.cornell.edu/uscode/text/8/1185)
This is a lot easier if you travel under the same name. However, if you are travelling to Russia, you are required to show your Russian passport as well. (http://stpetersburg.usconsulate.gov/dual-nationality.html)
If you have time, you may want to bring documentation of your name change and apply for an update to your Russian passport, or at the least, carry such copy with you. (I've used apostilles for this without problem in Czesko.) This happens to people for obvious reasons (marriage :-)) and they have procedures to cover it. At the worst, you may have to wait some extra time at Russian entry/exit while they verify your documents.
If you're not traveling to Russia, then there should be no problem whatsoever.