The Greeks might not be as unruly as we might assume. After all, Athens is likely destroyed, and Sparta is likely dismembered (an invading Persian army would surely free the helots, and re-found Messene). Thebes surrendered to the Persians IOTL (with the other Greeks remembering them as traitors), and Argos was neutral, and would have supported Persia had their victory become inevitable. Corinth is the strongest potentially unruly population, but it probably would have found that the empire was conducive to trade. If we look at the Macedonian occupation of Greece as a model, the smaller Greek states would probably love the Persians, as Persia could protect them from the antics of the big cities. And, as the Persians were pretty lenient to conquered peoples, and didn't interfere often in local government (surely they wouldn't as much as the Macedonians later would!), there is a chance that the Greeks might be more peaceful subjects than we assume.
Plus the Greeks have something that most of the other peoples (the Phoenicians being the main exception) that the Persians conquered don't: western colonies. This does allow a sort of "valve" for the strongest dissidents to escape to. A greater Greek westward migration might help Syracuse defeat Carthage in Sicily - the balance of those wars can be easily tipped. Perhaps a Syracusan tyrant could be the one who tries to liberate the Greeks from Persian rule?
Macedonia as a world power is likely butterflied. It's OTL rise is far from inevitable; in fact, it's really kind of weird. It was kind of an ancient Prussia, with Philip II being Frederick William I, and Alexander the Great being Frederick the Great. Although, then again, the rise of most historical world powers is kind of unlikely.
Greek culture should do well under Persian rule, and it should influence Persia. Since Athens won't flourish in this timeline, however, it probably won't do as well - at least, we'll be missing a few of the more famous thinkers. Other Greek cities were plenty capable of having their own philosophers and playwrights though. It might inspire the Achaemenids to use Greek hoplite tactics more than they did IOTL.
A conquest of Greece might help make Persia more stable than it was IOTL. After all, the Greeks helped encourage the Ionians and Egyptians to revolt, and without Greeks to fight, the Persians could pour more resources into stablizing other places. Then again, the Sicilian Greeks might just take the place of the main Greeks ITTL, encouraging both Greece (and Egypt) to revolt, which would be even more of a mess for the Persians to figure out. So I'm not sure how this is handled. In addition, we should see more Persian influences on Greek culture and philosophy, which would be very interesting in and of itself.
Rome could rise as it did IOTL, or that could be butterflied. If the Sicilian Greeks are stronger than they were IOTL, that could stifle Roman expansion. Maybe a longer-lasting Rome-Carthage alliance?
If/when Persia breaks up, a Persian occupation of Greece could make the process of unification for Greece much quicker. Who knows who unifies it, but a unified Greece would be a very powerful successor state, for obvious reasons.
I think this is one of the most interesting PODs there is. It's challenging, but I don't think it's as unanswerable as people think. I would do it myself if I didn't have other projects... maybe a TLIAD....