International Driver's License: Yes, no, maybe?
With a rental, your American license will get you through the day, the week, and maybe the month (remember that most rentals are stick shifts). That said, an International License from the AAA supports the validity of your U.S. license. If anything untoward happens, having one is helpful. Sometimes (as in all nations) it depends on the mood of those who stop you. The International License is a good faith, law-abiding gesture that keeps you on the more honest side of things, though honesty isn't necessarily the lay of this land.
If you intend to stay in Italy for a longer period, you may have to err on the side of caution and apply for an Italian permit, which requires a test and a learner's permit. It's an arduous procedure that most people tackle by attending a driving school.
Bottom line, driving in Italy on a North American permit only is illegal. Will police arrest you? Unlikely. But the letter of the law is a local permit supported by an International License or an Italian permit. Assuming your U.S. license will get you out of trouble or that playing dumb is a foolproof option can cost you dearly.