Voting
And one more thing....
In the case where you are born a dual-citizen, I think you are safe, but for a naturalized citizen, the dual status is a bit more delicate, since the US requires you to swear to "adhere to the constitution" or something like that.
If a naturalized citizen votes in elections in the "old country", the US can take that as an indication that they have not really become attached to the US. I'm not sure of the case, but I have a foggy memory of somebody having their citizenship revoked because they voted in a Mexican election. How the INS found out, or under what circumstances they raised their heads above their papers to do something, I don't know.
Most people figure out a way for their sons to avoid military service in the "old country". My friends are Russian. They had to file a special paper to indicate that their son lives abroad and will not be subject to service in the Russian military. Since Russian military service is universal and obligatory, this is a big deal. Without that, they might have either arranged for him to give up his Russian citizenship, or he simply would stay out of the country until he was 26 and could no longer be called.
-Ocelot