Well, if you decide to get a SIM at Termini, there are Wind and "3" shops there. The nearest Voda shop seems to be about a block away.

mobile carriers on termini website
 

Termini rail station has a website with some stores represented, including TIM, WIND, 3. Link is here. Drill down and there is some contact info and map with store location within the station. There is also an APP in the Android/iPhone app stores for Termini, which has maps etc.

Two years ago, I had a 5-hour layover between my flight from the US and my connecting flight to Palermo.

I believe I landed in T3 (arriving on Alitalia from LAX) and I was able to walk all the way to T1, where I saw the TIM and 3 shops. There was also a Dixons but they didn't have SIMs. My flight to Palermo boarded from near there, so I had to go there anyways.

I'm going to try to do the same thing again, arriving from LAX on Alitalia 621 later this month, then a 5-hour layover until a flight to Verona. So good a time as any to get the SIM and start using it while I wait.

Just came back from my trip and can now answer this question accurately. As of mid April 2013, getting to the TIM or Vodafone stores is impossible if you are arriving at Terminal 3 from an international flight. After arriving, I was routed through passport control and immediately was routed to baggage claim, even though I didn't have any checked bags.

I ended up getting a SIM card from a TIM shop on Via Cavour, pretty safe area, close to Termini, and the whole transaction took less than 5 minutes. Didn't even need a tax code and the SIM worked within 5 minutes. From my experience, there are TIM shops on the major shopping streets (Via Del Corso in Rome, Via Del Calzaiouli in Florence, etc) in every city. I paid 25 Euro for the SIM card, which included 1GB of data and 15 Euro worth of talk time. For my 3 week trip, this was plenty. Using Google Maps, I never get lost anywhere, even on the narrow confusing streets of Venice.

By the way, I also highly recommend bringing 2 batteries with a separate battery charger. I bought

Amazon.com: EC TECHNOLOGY 2X 1900mAh Li-ion Battery for HTC Shooter, HTC Sensation, Sensation XE, Sensation XL, HTC Z710E, Z715E, HTC EVO 3D, HTC EVO 3D(4G), MY TOUCH 4G SLIDE , AMAZE 4G & 1X Multi-purpose USB Travel Charger: Cell Phones & Accessorie

With this approach, I never even came close to running out of power, even after using navigation extensively from 8am to 10pm.

What if you're connecting through FCO, rather than staying in Rome?