We know a great deal about the phonology of some forms of ancient Greek. Part of our knowledge comes from ancient treatises laying out in great detail exactly how each letter was pronounced; part comes from analysis of spelling variations in ancient examples of writing; and part comes from out understanding of linguistic necessity and of what kinds of changes in language over time are possible or impossible.

We know a great deal about certain early dialects, especially about linguistic developments in the 6th and 5th centuries BCE, thanks mainly to epigraphic evidence: inscriptions that date to that time. This is because there were no universal spelling conventions at the time, and every state had its own alphabet and its own orthographic practices: the variations tell us a great deal. For example, variation between α, ε, η, and an aspirant character representing /h/ tells us a great deal about how different dialects dealt with the development of the early sounds /e/, /ɛ/, and /æ/ and how their phonology developed until they were eventually universally written as either ε or η. To be sure there are areas of uncertainty: we don't know how ττ was pronounced in Attic, for example.

Standardised orthography makes that kind of analysis considerably harder, and that's why we actually know a good deal less after the cut-off point of 323 BCE. From isolated pieces of evidence it's possible to pick up changes here and there, from widely separated time frames, but the timeline of when these changes happened is very much up in the air. We know the pronunciation of β shifted from /b/ to /v/, but we don't know exactly when. The shift of ε, η, ι, ει, and οι towards /i/ happened over a period of many centuries, but our timing of when each letter or letter combination shifted that way is very imprecise -- sometimes we can't put it any nearer than the nearest half-millennium.

The situation is really very similar with Latin, except that we only know about the phonology of one, or perhaps two, varieties of Latin, rather than the multiple models we have for early Greek dialects. The dating of the pronunciation practices we do know about is again very uncertain: it's likely that the Latin pronunciation taught as "standard" was already out of use, or at least antiquated, by Cicero's time. Latin orthography was never as standardised as Greek koinē, so clues do pop up now and then: we can get a lot of phonological clues from Plautus, and a lot from late antiquity, but in between there's not much and we can only fill in the gaps by positing intermediate stages that aren't actually attested. There are rare exceptions: one famous example is a well-known figure in 1st century BCE Rome changing the spelling of his name from "Claudius" to "Clodius".

But it baffles me why people who teach Latin or Greek would ever claim that we know nothing, or even not very much, about the phonology of these languages. Phonology isn't the same thing as pronunciation, to be sure, but when people talk about "the pronunciation of Latin" it's usually the phonology they're interested in, even if they call it "pronunciation". Sure there are gaps in our knowledge, but really we know an extraordinary amount. Well, perhaps the problem lies in the "we": "we" is really a fairly small group of experts. To begin a journey towards membership of that group, for further reading I'd recommend Sidney Allen's books Vox graeca and Vox latina, and (more up-to-date, and differently organised) Horrocks' Greek: A History of the Language and Its Speakers.

If we know what ancient Greek sounded like, it has been figured out since I finished my classics major twenty-some years ago. There are standards of pronunciation to allow classicists to talk about the language and the texts written in it, and not everything is unknown, certainly. But no one supposes the modern conventions fully reflect the daily experience of the language's sound on the streets of an ancient Greek city.