I've been researching this quite a bit myself, though I am more interested in having internet connectivity while out and about so I can look up information like maps, reviews, etc. There are a couple of different Italian wireless carriers that offer 'no contract' plans. As long as your phone is unlocked and supports international travel, you should be able to pop out your US SIM card and replace it with one you purchase in Italy. I have an iPhone 6S that qualifies on both counts.
The service I am considering is TIM for Visitors. For 30 euro you get 4 GB of data and 100 minutes of voice calls to a large number of countries, including the U.S. You have to purchase online from the U.S. before you go. Supposedly you get an email confirmation which you take to any TIM store in Italy (of which there are many, I'm told) and they will install the TIM SIM for you. Since I haven't yet done this (I'm going next week) I can't report on whether this is a good option, but all of the research I have seen tells me that this is a popular and sound choice. https://www.tim.it/tim-visitors-en
That said, I have seen a few cautionary tales, namely that not all TIM stores are created equal. To be safe, people suggest printing out the promo page in both Italian and English. That way when you show up at the store if the clerk is confused or wants to charge you extra for the SIM (which is generally 10 euro, independent of the plan) you can show them the details in Italian. I don't believe you can access this deal within Italy, so if you try to show the webpage to the clerk on one of their devices, it won't show up.
The other drawback is that you can't text with this plan. That is easily remedied though by installing Facebook Messenger, What'sApp or some other text messaging app. I also think if you have an iPhone, iMessage will work. In addition, you will be assigned an Italian phone number. Not a huge deal, but if you use an app like WhatsApp that's linked to your phone number you will need to let your contacts know what your Italian number is so they can create a temporary new contact for you.
The other local Italian carrier is Vodafone. I don't believe they have any special tourist plans, but much like the pay as you go carriers in the U.S. (Boost, Metro PCS, etc.) you should be able to buy a one month plan. You will just need to swap out your SIM card. The drawback to this is that you may need to physically cancel the service when your vacation is complete to avoid monthly charges. With the TIM SIM it is valid only for 30 days from the date of activation.
Good luck and have a wonderful time!