Many cultures with epic traditions do (and did) sing their epics with musical accompaniment, like the examples Tiako mentions, or the guslars that Parry and Lord recorded in Bosnia. In the Greco-Roman case that you're asking about, though, it's a different story: there it's a blend of historical reality and a metaphorical poetic conceit.
It worked differently depending on what poetic genre you're talking about. Lyric poetry was originally sung with instrumental accompaniment, but as time passed this came to be more a metaphor than a reality. In the case of hexameter poetry - which includes epic - the reverse may well have been the case: it seems to have started off recited, not sung, but there were lots of metaphors describing the act of performing as "singing"; as time passed, sometimes people took the metaphors literally. As a result musical performances could be put on, though only for the "classics" of the genre (Homer), and only occasionally. In Vergil's time poetry in both genres was normally recited, not sung, but special performances could be musical, like Nero's semi-legendary performance of the Sack of Ilium accompanying himself on the lyre. Lyric poets like Horace were not singers, though Horace uses loads of singing and lyre metaphors throughout his poems.
The case of early Greek epic is the trickiest one to handle, because there's such a shortage of evidence on how it was performed. The existence of parallels like the Bosnian guslars, and in the epics themselves the depictions of poets as singers, along with the pervasive use of singing metaphors, creates a widespread assumption that Greek epic was sung. There's a lot of inconsistency with that picture, however. There's probably room for disagreement, but in my reading the upshot is that it is not a realistic picture.
First, realise that we have two types of testimony about the performance of early epic: (1) in the poems themselves, poets are described as aoidoi "singers", who sing (aeid-) and use a lyre; (2) other ancient testimony refers to performers as rhapsodes, and rhapsodes did not sing, but instead recited while beating time with a staff.
In the Hesiodic Theogony, lines 30-1 the poet describes himself as holding a staff, i.e. he is acting the part of a rhapsode. A few lines later, he refers to the act of performance as "singing" (aeidein). These are mutually exclusive, so one of them must be metaphorical.
Pindar Nemean 2.2 refers to epic performance by "singers of stitched hexameters". "Singers" implies musical performance; "stitched", in Greek rhaptoi, is equally explicit and refers to rhapsodes. Again these are mutually exclusive, so one must be metaphorical.
Plato (Ion 553c), talking about the Odyssey character Phemius, refers to him as a rhapsode even though the Odyssey has him using a lyre. Same point again.
Numerous sources refer to how early hymnic poets, especially Olympus and Terpander, sung hymns (in hexameter, like epic) to music; they were unquestionably musicians, and they're linked to all sorts of innovations in music. But this is a separate genre: the sources refer to hymns very specifically, and these hymns must not be conflated with epic, or with the poems that served as preludes to an epic performance (like the Homeric Hymns). The surviving fragments of Terpander are also distinguished from virtually all other hexameter poetry, including epic and the Homeric Hymns, by the fact that they are not in the Ionic dialect.
We have one extant musical score for a hexameter poem. But it is late (2nd century BC), and it is a religious hymn (to Asclepius). Again, religious hymns cannot be conflated with hymnic preludes.
Heracleides of Pontus (fr. 157) tells us that one of Terpander's musical innovations was setting Homeric poetry to music for the first time.
We have one pictorial depiction of someone performing a hexameter poem, and it is a rhapsode, not a singer (a vase by the Cleophrades Painter, Brit. Mus. E270; the other side of the vase shows a musician, but he is not accompanying the rhapsode: (a) in epic, aoidoi accompany themselves, and (b) he is playing an aulos, not the conventional lyre).
Altogether, these points suggest that even in Homer, the idea of describing epic performance as a musical performance is a poetic conceit, a conventional metaphor rather than a reality, just as it was for Vergil.