Customs time - Fiumicino to US

I'd take an earlier train out of LaSpezia. Easier to leave earlier than miss your flight home.

You have no buffer. It is too tight by at least two hours. At 3 pm it is most likely a direct flight back to the US which means that you have to first use Terminal 5 for departure. At that time of day, Terminal 5 can be a mess with long lines. Second, the recommendation for check-in is three hours even with carry on. Third, it not unusual for check-in to be cut off 45 minutes prior to departure which means you must be through check-in by 2:15 and standing in-line at 2:15 does not count. Roll everything forward two hours. Finally, you timeline works only if everything else works with no errors or delays.

Wow..thanks for your responses! I'm glad I asked.

Could you please specify your flight number?
There is only one flight departing directly to the US after 1pm, and that is Alitalia AZ610 (Delta codeshare DL1043) to JFK, currently departing at 14:45. IF that is your flight, the check in is at terminal 3, since it's operated by Alitalia. You must have completed the check in procedures, I.e. have the boarding pass and baggage check claim in your hands 45 min prior to departure. Being there 2 hours prior to departure is generally sufficient, assuming you have no delays with your La Spezia train.
However there is also the possibility that you might be on another flight stopping first in some other other EU hub. For example I see BA flying to LHR at around 15:00. That's why I'd like to know your exact flight info. If you go somewhere else in Europe first, then it's ok to get to the airport 75 to 90 min prior to departure.

my flight info is:

Monday, May 19

Flight: Alitalia 628
Departs: 3:00 pm from Rome, Italy
Arrives: 6:40 pm at Chicago - O'Hare, Illinois

I forgot about the seasonal (summer only) Alitalia flight to ORD.
It's term. 3, right in front of the train station.
The rule is what I told you. You must have completed the check in 45 min. prior to departure. I would say that being at the airport 2 hrs prior to departure is generally sufficient. Also that is one of the last Alitalia flights out of Term 3 (International Terminal) therefore there shouldn't be the zoo you have early in the morning when most flights to North America depart.
All European flights depart from terminal 1.

Even though your original plan can get you to the airport with two hours to check in, I would leave earlier to build in time for anything that might go wrong. Missing your flight would usually mean having to buy an expensive new ticket. For instance, take the 6:51 departure from La Spezia that should get you to Fiumincino by 10:52, including 17 minutes for the connection at Roma Termini. (If you're leaving from Vernazza, you leave there at 5:00.) The next departures that by DB shows from La Spezia have three connections, making them less attractive.

Thank you again for the advice everyone. We'll be taking the 06:51 out of La Spezia just to be safe. It means we have to cut short one night in Riomagiorre and stay in La Spezia the last night, but we'll at least get to spend most of our last day in Riomagiorre. Can you tell this is our first time travelling overseas? :)

Cheers!

I don't know how important spending a last day in Riomaggiore is to you. But I like to spend the night before my flight in or in the vicinity of the city I'm departing from. In you shoes I would probably spend the last night in Rome. As I said 2 hours prior to take off should be enough, but I don't like to stress over the possibility something could go wrong with the train journey.