Most definitely true, but you're right to suspect that different cultures had influence at different times. The Greeks had a huge influence in early Rome, since they had colonized Southern Italy. You probably heard in your class about how one of the best representations of Greek pottery, the Euphronios Krater, was found in Italy. And then there are the other peoples who were in Italy during the rise of Rome...my personal favorites are the Etruscans, but there were a lot of other peoples in ancient Italy. The Greeks had one of the most lasting influences...you can see it in the poetry (Catullus is probably the earliest example we have, although the plays of Plautus came from Greek models, too), and the Romans always had a complex about not being as good as the Greeks in matters of art.

However, moving to the empire, you see an exchange of influence develop. A really good visual representation of this is the Fayum mummy portraits, which feature Roman citizens mummified in the Egyptian style, but with Greek inscriptions and Roman-style painting. Let me know if you'd like other examples: I just taught a Roman archaeology class and I'm full of them!

I don't really agree with your professor that there's no distinctive Roman culture: they definitely had a unique perception of themselves, and what it meant to be Roman. You can see it very clearly in authors like Cicero and Virgil.