The only airline with a 777 service to Rome from the USA is Alitalia, from JFK
and Newark (I assum, you are trying to go from the NY area). Delta flies the 767-300, Continental the 767-400, and American has a seasonal nonstop from JFK
on a two-class 767-300.
Personally, I would avoid AZ
. The service is surly at best and unreliable. The airline is going through a painful reorganization and staff strikes and work actions are common. The cabin crew are professional but rude and disinterested. The 777's are supposedly lovely, but not worth the headache factor.
I'd go with either Continental from Newark or US Airways from Philadelphia, which flies the A330 on the route.