That is a very broad generalization. In many jurisdictions, including the one where my wife has served as an election judge, if there is any doubt a voter is not a citizen, he or she is given a provisional ballot. There are something like 3900 boards of election in the U.S. Not all of them are sloppy about the process. As you might have noted, several states have passed legislation mandating the showing of identification before voting, and the Supreme Court has ruled such legislation to be constitutional.
If Senator Obama were not an American citizen, wouldn't his political opponent have raised the issue in Illinois? After all, this is not the first election Obama has stood for.