Writing "Ancient Roman" music is problematic because the Romans left us very little notated music. The surviving music uses a neumatic, or differential, system of notation, in which only changes in pitch relative to the previous note are represented, rather than absolute pitches. Scholars can get a general sense of the shapes of Roman melodies, although they are unsure which side of the page is the top. Rhythms are unclear.
While Coptic hymns can serve as a close approximation of ancient Egyptian music, early Catholic hymns cannot be used to approximate Roman music. The early Christians (being an oppressed minority) sought to differentiate their music from that of the Romans. However, this does suggest that four measure phrases, the scales common in Gregorian chant, and other Christian musical devices be avoided for a more authentic sound.
As in most cultures, vocal music was very common. And in this case we know the language. Furthermore, there is a wealth of Latin poetry. So vocal music using Latin texts is a possibility.
The poet Ovid's Fasti (Festivals) can be found in Latin here: http://www.thelatinlibrary.com/ovid/ovid.fasti4.shtml
, and in English here: http://www.poetryintranslation.com/PITBR/Latin/Fastihome.htm