The Greek mainland as a whole did not rely on imported grain. Only Athens did. Most Greek states pursued an ideal of self-sufficiency: they tried to make sure that their own territory supplied all their basic needs, and looked to trade only to import exotic luxuries. While coastal areas might be able to respond to shortages by tapping into a wider Mediterranean grain market, many states were not even usefully connected to such a network. Importing enough grain to feed the population would simply cost too much.

Therefore, while Persia could have tried to stop the grain export from the Crimea - and the easiest way to do it would have been to blockade the Hellespont, as the Persians actually did with their pontoon bridge in 480 BC - the only Greek state substantially affected would be Athens. Other states might feel the squeeze if there was a local drought and no Black Sea grain to fall back on, but in a normal year they would not need to import any, and if they wanted to they could easily turn to their settlements in Sicily for help. Sparta and the rest of the Peloponnese could continue resisting Persia indefinitely unless the region was subdued by force.

And even then, it's not likely that Athens would be "destroyed" by such a blockade. It's true that they relied very heavily on Black Sea grain later, and that much of their strategy during the century after the Persian invasions was focused on keeping the Hellespont open. But that doesn't mean there were no other ways to import food. Admittedly a lot of other major grain-producing regions (like Egypt and Cyrene) were also nominally under Persian control, but there were other places exporting grain (like Sicily), and besides, it would certainly not be extraordinary if Athens deliberately fomented an uprising in Egypt just to secure a supply of grain. In any case, part of their obsession with keeping control of the Hellespont was that the Athenians had claimed and settled the Thracian Chersonese (modern Gallipoli), which also exported grain back home.

I'm leaving aside here the very likely scenario in which Athens would go to extreme lengths to reopen the Black Sea grain trade. If the Persians tried to enforce a naval blockade or to occupy the Greek states on the Crimean Bosporus, Athens would probably respond with military force, and would find willing allies among both local Greeks and neighbouring Skythians. But this depends on what form the Persian attempt at an economic embargo would take, and of course it's impossible to predict how such a conflict would go.

In short, Persia could have used economic sanctions only against Athens; it wouldn't have been very effective and would likely have prompted a massive military response. It's hard to see how Persia would gain anything from such a strategy. They never meant to "destroy" Old Greece anyway, but rather secure its (ideally willing) submission to Persia and the resulting tribute payments. If it were actually possible to starve Greece to destruction, would there be much left to gain from taking control of the area?