The US is fine with dual-citizenship, it just doesn't encourage it because it can make certain policy issues a bit more awkward.
Sources:
".U.S. law does not mention dual nationality or require a person to choose one citizenship or another. Also, a person who is automatically granted another citizenship does not risk losing U.S. citizenship. However, a person who acquires a foreign citizenship by applying for it may lose U.S. citizenship. In order to lose U.S. citizenship, the law requires that the person must apply for the foreign citizenship voluntarily, by free choice, and with the intention to give up U.S. citizenship." [1]
"Dual nationality can also occur when a person is naturalized in a foreign state without intending to relinquish U.S. nationality and is thereafter found not to have lost U.S. citizenship: the individual consequently may possess dual nationality. While recognizing the existence of dual nationality and permitting Americans to have other nationalities, the U.S. Government also recognizes the problems which it may cause." [2]
Good reference. I have been with friends in years past for their U.S. citizenship swearing in. I recall they had to turn in any other passports they had (at least they were told they had to). I guess I'm still confused.
This doesn't sound right. My wife was just recently sworn in and was not asked for her previous passport, and one of my daughters just got her citizenship, and was not asked for her passport either.
Sources:
".U.S. law does not mention dual nationality or require a person to choose one citizenship or another. Also, a person who is automatically granted another citizenship does not risk losing U.S. citizenship. However, a person who acquires a foreign citizenship by applying for it may lose U.S. citizenship. In order to lose U.S. citizenship, the law requires that the person must apply for the foreign citizenship voluntarily, by free choice, and with the intention to give up U.S. citizenship." [1]
"Dual nationality can also occur when a person is naturalized in a foreign state without intending to relinquish U.S. nationality and is thereafter found not to have lost U.S. citizenship: the individual consequently may possess dual nationality. While recognizing the existence of dual nationality and permitting Americans to have other nationalities, the U.S. Government also recognizes the problems which it may cause." [2]
[1]: http://travel.state.gov/travel/cis_pa_tw/cis/cis_1753.html
[2]: http://travel.state.gov/law/citizenship/citizenship_778.html