Michael,
I can assure you that there ARE many rail solutions on that route. To begin with, forget using Eurail for researching any rail trips. For trips in Italy, the best two websites are Trenitalia or bahn.de.
As someone else mentioned, it's far too early for schedules to be posted for May 2016. Rail schedules are loaded twice yearly and I believe the next update will be in December. You can get a good idea on availability on that route by entering a date within the next week, on the same day-of-the-week that you'll be travelling. Schedules may change slightly next year, but probably not much.
On that route you'll have either one or two changes (most likely including one at La Spezia Centrale), and will use a combination of Regionale, Freccia and InterCity trains, depending on which departure you choose. When you buy tickets, you'll need to specify which of the five towns you want as the destination.
As the others have mentioned, Rail passes are rarely a cost effective method for travel in Italy as trains are relatively inexpensive (especially the Regionale trains). One other point to note is that Rail passes do NOT include the seat reservation fees which are compulsory on the high speed trains. Pass holders must pay separately out-of-pocket for those, and the reservations are specific to train, date and departure time so can only be used on one specific train. Anyone caught without valid reservations for the train they're riding on will face hefty fines, which will be collected on the spot (the same applies for those travelling on Regionale trains with unvalidated tickets).
On the trip from La Spezia to Roma Termini, I'd suggest using one of the direct departures, of which there are several each day. Those provide a travel time of about 3H:45M and no changes. It's a very pleasant trip.