Updated Aug 2017
I live in the US but travel to Italy 3 to 4 times per year; here are my suggestions on how to use a US mobile phone in Italy:
Short stays (0 to 4 weeks)
I switched to Tmobile as a carrier 2 years ago when they started offering very good deals for overseas travelers.
Tmobile offers free text messages, free email, and free 2G connection for web browsing and other applications for travelers to 140 countries, including Italy. I personally use an iPhone (but any smart phone will do) since it switches to wifi calling, rerouting all calls to the internet when wifi is available.
If wifi is available, then I can make and receive phone calls for free using my US phone number, just like I was back in the States. If wifi is not available, Tmobile charges 20 cents a minute to use the cellular network. That is not much of a charge, but if you call often you can install an application like WhatsApp.
The free 2G speed (128kbs) that Tmobile offers is enough to use WhatsApp for free and call the US using the App, so my phone calls costs are not significant anymore when I travel. Also the 2G speed is good enough to use Waze, Google maps or any other phone navigator.
In 2017 T-mobile added a new service T-mobile One Plus and if travel frequently internationally and require a higher speed you can add T-Mobile One Plus™ and get up to 256kbps speeds abroad for $10/mo.
If you use another carrier, you should check your carrier deals.
Long stays ( 4+ weeks)
If you wish to stay in Italy for a long time, the best solution is to buy a local SIM card. Italian carriers like Tim, Wind, Tre offer a card; the problem is that, when you purchase one, they ask for a document and an Italian "codice fiscale" (sort of a Social Securty Number). Since I have an Italian Codice Fiscale it was no problem for me to get the card, but when you don't have one, you may need to ask an Italian friend to buy the SIM for you.
By Paolo Nascimbeni