Short answer: we have no idea.

Long answer: while there are instruments, and notated music that survive from that time, there's no direct correlation between Roman written music and anything that survives. No Rosetta Stone-style document that allows us to interpret what was written into what was performed.

That being said, there's lots of educated guesswork. As I said, there are surviving instruments, and depictions of how they were played. But how they were tuned, strung, and played is somewhat of a mystery. To give you an example as to why, imagine I were playing a flute, and I commissioned a painter or sculptor to depict me playing the flute. Would they include all the holes in the right spot? Would they get my hand position right? Would they have my mouth in the correct position, or even have the hands on the right side of the instrument? So yeah, there's a lot of unknowns.

We know there were these instruments, and we can make educated guesses as to the interpretation of notation, using things like Byzantine music notation as a framework and extrapolation of Greek notations, and using some assumptions about use of the Pentatonic scale, approximate pitches and so on.

So, with all that guesswork, you end up with recordings like this: https://www.youtube.com/watch?v=uJLXyBzMci0&list=RDuJLXyBzMci0#t=0

which, while they're as accurate as the educated guesswork and interpretation/extrapolation can get us, are still just an interpretation of what we think Roman music sounded like.