I'm wondering if those ancient stories were still told in the land where they originated into the Middle Ages, or if they fell out of favor or were suppressed by the Christian Byzantines.
According to Judith Herrin’s Byzantium and Lost to the West, both epics alongside many other examples of Greek literature was studied and told throughout the Middle Ages. While it mainly concentrated around the educated in Constantinople and Mystras (as the fortunes of the Empire wanes), there was one instance where an educated noble quoted from the Iliad in order to woo a noblewoman. However, since she wasn’t educated, she asked for it to be interpreted. Greek philosophers were often depicted in mosaic form and in the capital were studied by their intellectuals. The Library of Constantinople held many of the classics aside from other books that were later taken West in 1204 during the 4th Crusade.
Another instance comes from the history of Anna Komnena, a notable female noble and scholar, who would’ve learned the classics. As I noted above, a woman being educated, especially a noble woman, would have been the exception rather than the rule in the Roman Empire of the medieval period.
In the Lost City of Byzantium, Mystras is credited as the progenitor of the Paleologan Renaissance, a revival of Hellenistic thought which throughout the decline of the Byzantine Empire, gradually immigrated to Italy, forming schools in each of the city states where philosophy, the arts and other ideas of the Hellenistic world permeated back into Western thought.
As an interesting thought,some intellectuals in the Empire during the Empire’s final years believed in a return to Greco paganism if not reforming the Empire as a Greek state, showing the beginnings of Greco Nationalism at the waning days of the Empire in the 15th century.