Neeraj Mathur is quite right: syllables, not letters. Some circumstantial evidence for this from Ancient Greek drama. When literacy was a very new thing, and the tools of grammatical analysis (such as words) were still not very popular. (https://www.quora.com/Could-sayi…):
the differentiation between utterance and word was newfangled with the Greek sophists. Aeschylus avoids Euripides’ new fangled distinction in Aristophanes’ Frogs, and sticks with the time-honoured epea, “utterances”—which could be as short as an ouch, and as long as the Iliad.
At the start of Aristophanes’ Knights, Nicias asks Demosthenes to keep repeating molōmen and auto really fast. That makes a pun: molōmen and auto “let’s go (to) it” turns into auto molōmen “let’s run away” (somewhat taboo; more taboo with the characters saying molōmen auto molōmen auto molōmen auto molōmen to the rhythm of jerking off). That displays an awareness of words being part of other words, but not of phonemes.
For “say it quickly”, Nicias says xyllabōn. In non-Attic, that’s syllabōn: “taking it together”. Something that you take together—like, say, a bunch of phonemes, or I’d say, a sound that you take together as a single quick unit—is a syllabē: a syllable.
And the old-timer Aeschylus, who did not want to know about Euripides’ newfangled words, does speak of syllables, and graphemes, and NOT phonemes. Seven Against Thebes, 468:
“This one, too, shouts in syllables of written letters (grammatōn en syllabais) that even Ares could not hurl him from the battlements.”
Letters in Greek are literally “written things”: grammata. You shout in syllables. Aeschylus knows that multiple grammata correspond to a syllable. But he hasn’t quite made the leap to one gramma corresponding to one phoneme—even though that’s why distinct letters exist to begin with. Even if you interpret syllabē as “bundle”, you don’t sound letters out separately.
Similarly, Greek had a word syllabizō for pronouncing the syllables of a word one by one, used by Plutarch. The corresponding word grammatizō, used by the grammarian Herodian, doesn’t mean pronouncing the phonemes of a word one by one: grammata are “written letters”, and grammatizō means “to spell”.
Plato, it’s fair to say, uses phōnē “sound” to refer to vowel sounds as distinct from consonants. So there was the beginnings of an awareness of phonemes in Greece; after all, their alphabet was phonemic. But my hunch is still that it would be Zer-kes’ vs Zer-ges’ new cave, not Z-e-r-k-e-s, that syllabising rather than phoneme-ing was the default.
Btw, in Modern Greek, syllabising combines with phoneme-by-phoneme spelling. The informal Greek alphabet is vowels, and consonants followed by /u/. So Zu-e-ru-ku-e-su (instead of Zeta Epsilon Rho Kappa Epsilon Sigma).