It's possible that some Greeks "maintained" Classical naming traditions through the Byzantine and Ottoman periods, but this is largely a relatively recent phenomenon. Of course, by "recent" I mean eighteenth-century—a time when Ottoman Greeks, and especially members of Greek diasporic communities outside the Ottoman Empire, "rediscovered" their classical heritage with the encouragement of European Philhellenes. Byzantine Greeks had maintained an sense of continuity with their Classical predecessors, but that was largely lost in the Ottoman period. Ottoman Greeks' sense of their past, especially the historical model endorsed by the Church, tended instead to emphasize the nation's Byzantine, Christian past. (ETA: both of the linked answers are by me, pls no kill, mods!)

The adoption of names from antiquity, and therefore entirely devoid of Christian associations, came as something of a nasty shock to the Church! An encyclical issued by Patriarch Grigorios V of Constantinople in 1819 roundly condemned the practice:

And the innovation introduced, as we have heard, of giving ancient Greek names to the baptised infants of the faithful, taken as a despising of the Christian practice of naming, is altogether inappropriate and unsuitable. Therefore it is necessary that Your Reverences give strong commands to the priests of your parishes, and spiritual admonitions to your blessed parishioners, to abandon forthwith this abuse [ ... ] and parents and godparents in future are to name at the time of the holy and secret rebirth with the traditional Christian names, to which pious parents are accustomed, the [names] known in Church, and of the glorious saints that are celebrated by it, so as to be overseers and guardians of the baptised infants [ ... ]

Nor did the trend stop at parents naming their children; enthusiastic young Greeks would occasionally rename themselves in antique style. The French traveler Ambroise-Firmin Didot records one such group as well as the text of the resolution they adopted (it includes a Themistocles for you!):

Resolution

Under the direction of Theophilos, Grigorios and Efstratios, teachers in the Academy (Ellinomouseion) of Kydonies, on the 20 March 1817, the undersigned have resolved:

Wishing to take up our ancestral tongue, and ardently desiring to reject the gross and vulgar language, as wholly unbecoming to us the descendants of those Hellenes, we have all decided to decree this law, so that each of us, whenever we gather together, is obliged to converse in the Hellenic tongue [ancient Greek] .

Law

Each of us is to speak, as far as possible, in the Hellenic language. Whoever does not do this, is, as a punishment, to recite a page of Homer before us.

Those resolving

Angelis ... Alcibiades / Samouil ... Nicias

Didot ... Anacharsis / Ilarion ... Xenophon

loannikios ... Aristeides / Panayiotis B ... Pelopidas

Tzanos ... Epaminondas / Ioannis ... Pericles

Dimitrios ... Themistocles / Leontios ... Phocion

Theophanis ... Cleanthes / Konstantinos ... Chabrias

Dimitrios K. ... Miltiades / Kharalambis ... Pausanias

Vasileios ... Agesilaos / Georgios ... Patrocles

Methodios scripsit ... Diogenes

The "Hellenic language" here means Ancient Greek, as opposed to the "gross and vulgar language" generally spoken by the group and their contemporaries. The language tension here—a reflection of the cultural debate over whether the state should adopt a "purified" and archaic language (katharevousa) or the vernacular (dimotiki) as its official language—mirrors the name issue in many ways. Of course, both purists and demoticists were working within a general framework of Hellenism—by this point, the idea that ancient Greece should serve as a model for the modern state was a commonplace. But the question of how far Greeks should go in imitating the ancient past was—and, if a bit less so, still is today—a hotly debated question.

[All documents quoted are from Richard Clogg, ed., The Movement for Greek Independence, 1770-1821: A Collection of Documents (Macmillan, 1976).]