Although there are suggestions that the Romans borrowed the Greek notation system we don't have any complete pieces of music written in that from Rome to work with (that I know of anyway) and I think there's only one from Greece!
However there is a group of musicians/archaeologists called "Synaulia" that have spent a large amount of time attempting to research this very question and having used iconography, literary references and various forms of cultural study whose names my brain is refusing to divulge right now first managed to replicate a variety of instruments that would have been in use during Roman times (including those borrowed from the Greeks) and used them to record what they term "a hypothetical reconstruction of the music of ancient Imperial Rome".
So being generous this falls under the heading of "a best guess" but they have released two albums of their recordings titled "La musica dell'antica Roma" (literally "The Music of Ancient Rome") vol. I and vol. II
The first volume focuses on the Roman-era wind instruments and the second is strings.
In the absence of any concrete archaeological finds of transcribed music this is probably the closest we'll get I'm afraid.