Tonight I will be leaving Rome to go to Boston, with a stop in Frankfurt. Both legs are operated by Lufthansa, and I already checked-in online and have both boarding passes, but will my checked-in baggage be directly delivered to Boston after I leave it at a drop-off counter or will I have to check it in again at Frankfurt?
Based on what you've said, you are simply flying two legs on the same airline, booked on the same ticket.
Given this, your luggage will be checked all the way through to Boston. In Frankfurt you will need to go through Passport Control (as you are leaving the Schengen region), however you will not need to collect your baggage - it will be automatically transferred for you.
It's always a good idea to check when you are dropping of your luggage that they do correctly label it as going to Boston - check the tag as they put it on the bag, and/or the receipt they give you for the bag and make sure that it mentioned BOS. In a case like yours (same airline, same ticket) it's very hard for them to make a mistake and get it wrong - but whilst there's humans involved there's always a chance!
Short answer: Almost certainly yes -- but double check with the check-in agent and then double check the final destination on the baggage stub you get. Longer answer follows.
If you booked both tickets together then definitely yes, it'll be booked as a single journey and your luggage will go to Boston as there's no policy requiring you to recheck luggage in Frankfurt (or inside the EU, that I know about). The only case were you might have to recheck the luggage is on a very long stopover (longer than 24 hours) some airports will not let you do this, but I suspect this is not the case for you.
If you've booked the tickets separately then you'll need to make clear to the agent on the check-in desk that you want the bag checked all the way, and show your passes. As long as the details match on both tickets they should do this. Now, I've had to insist that they do this in the past because their first position might be to say it's impossible, but it's not and they definitely can do it. That was on KLM but Lufthansa should be similar.
If you want backup, and you have separate tickets, contact customer services (contact details) and explain to them.