http://www.immigrationunitedstates.o...tizenship.html

As far as the U.S. is concerned most foreign aliens who become Americans can keep their original citizenship, however their country of origin may not allow double citizenship and they may have to give up their original citizenship. Therefore is very important to check with your country of birth if you can keep your citizenship when becoming American. It is important to consider the pros and cons about dual citizenship carefully before applying for U.S. citizenship.
Not quite, you are expected to swear loyalty to the US but you don't have to give up your former citizenship and can have double citizenship if your old country allows it.

Germany has relaxed the rules a bit, many Turkish people are allowed to keep dual citizenship and becoming a citizen of another EU country or Switzerland does not necessitate the loss of German citizenship anymore.