Oh, the most basic words you can imagine will change pronunciation radically over time. Just think how different English sounds today from how it sounded in Chaucer's time - let alone Anglo-Saxon times, when it was effectively a different language. And that's only going back half as far as the times of Alexander the Great. Even the English word "yes" changes pronunciation if you consider both the archaic "yea" (pronounced "yay") and the modern slang "yeah" (pronounced "yair"). (Compare the French "ouai".) In Anglo-Saxon, "yes" was "gea" or "gyse". So it's easy to see quite radical changes in pronunciation even to common words like that.
In the case of Greek, there were significant changes in pronunciation even within antiquity. β, for example, was pronounced "B" in Alexander's day, but by the time of the New Testament, it was pronounced "V", as it still is today. (Compare the change in Latin in the sound of V, which went from a W sound in classical times to V by the fourth or fifth century AD - which means that Julius Caesar really pronounced "veni, vidi, vici" as "weeny, weedy, weaky" - and the game has Caesar speak with correct classical pronunciation too, as far as I can tell.) What's more, quite apart from particular vowel or consonant sounds, ancient Greek would sound quite alien to any modern European, because it was accented completely differently. Modern European languages, including modern Greek, use stress to accent words. But ancient Greek used pitch. In that respect it would have sounded more like modern Japanese, which does the same thing. For these reasons, no-one today can really speak ancient Greek in a way that would sound completely accurate to someone from the fourth century BC; I think it's harder to get it authentically right than it is with Latin. But just as classical Latin wasn't pronounced anything like modern Italian, I don't think that classical Greek sounded much like modern Greek. In both cases it's easy to forget that, though, because ecclesiastical Latin is pronounced much like Italian (thanks to Italian dominance of the Catholic Church), and ancient Greek looks just like modern Greek, at least as it is traditionally printed.