The general assumption behind almost the entirety of the fantasy genre is that it is set in a pseudo-medieval period. Maybe this is so that the characters can wear pants, or maybe this is because it's how Tolkien did it, but I really started to question whether it was the right way to go some time ago. While I still enjoy fantasy books and gaming, I feel that the way to make it more interesting is to go Greek. (hint: thesis statement here-->) The reasons I think the Greek civilization is a better model for the fantasy genre is because it was polytheistic, it consisted mainly of independent city states, and because the monsters of its mythology are so commonly associated with the fantasy genre.
Starting with polytheism versus monotheism, many people are willing to ignore this in favor of capturing the overall feel of medieval Europe, but it really is kind of a big deal. In the middle ages, the Christian church dominated everything. For a thousand years, they were the dominant power. It really wasn't until Charlemagne came along that a secular leader managed to recapture real power from the church, and it definitely wasn't until the black death that people realized that the church didn't have all the answers. It wasn't until the crusades, the cultural exchange between the Muslims and the Christians, and the rediscovery of Aristotle that people in Europe realized that they could apply reason rather than faith as the primary means of making decisions. This is huge. It was so huge that the Thomas Aquinas spent a great deal of time and effort trying to merge the reason of Aristotle in with the dogma of religion.
Most fantasy genres, on the other hand, are very polytheistic. Having a multitude of gods that are constantly struggling against one another just makes for more interesting stories. It's more fun creating stories that involve mortals landing in the middle of struggles that are so much larger than they are. Polytheism gives characters an ultimate frontier to strive towards, and it also provides them with a reason to quest on behalf of their deities. In Star Trek V, Kirk asks "God" what he needs with a starship. It's a valid question in a universe where gods are all-powerful, all seeing, and all knowing. In a fantasy universe, the gods are a bit more limited in their abilities, and a Greek god would have a pat answer for Kirk: "Because Zeus got pissed, dumped me off here, and your starship is the only way to leave this dead planet!"
Religious matters aside, fantasy also usually assumes a degree of political intrigue. I'm not going to attempt to make the argument that the middle ages lacked politics - far from it - but I am going to say that a setting where city states were independent and controlled a great deal less territory makes for a more interesting story or game because the politics can change drastically as the heroes travel from one locale to the next. In a kingdom that spans hundreds of miles, if you make an enemy of the wrong person in one town, you will be wanted for most of the other towns within reach. In ancient Greece, if you piss off a king of one city, you make haste out of town, or hop a ship, and when you arrive at the next city, odds are that you get to start clean (or at least with whatever baggage you have from the last time you visited). It's easier to build an episodic story while maintaining the same basic culture throughout.
If you know your history, you could point out that Greece wasn't entirely without large government. There was the Delian League and the Peloponnesian League, which were comprised of a number of city states and were aligned against one another. This is true, and it's also true that Crete was a dominating city state during the bonze age. Despite this, the political influence of Athens, Sparta, and Crete was not as all-encompassing as it would be in a kingdom. The political institutions were completely unstable, which is why looking up the Delian League in Wikipedia turns up a huge list of city states that belonged to it at one point or another, but either dropped out on their own or were conquered by someone larger, forcing them out of the league. Many of the cities of Ionia, for instance, in Modern day Turkey, were part of the league until the Persian Empire came in and conquered them. Persia, in itself, makes for an excellent "evil empire" for characters to struggle against.
Finally, a good percentage of the monsters that form the backbone of fantasy fiction and gaming are either directly taken from Greek mythology, or they are based on these creatures. Off the top of my head, Greek mythology directly contributed the following monsters to D&D: minotaurs, cyclops, lamia, medusa, hydra, dragons, gorgons, centaurs, giants, hags, harpies, demons, ghosts, and more. Or, to put it another way, a good number of the monsters that characters run into in fantasy literature are directly derived from Greek mythology.
Finally, Greek mythology is more loaded with traditional fantasy heroes than medieval literature is. Sure, medieval literature has Beowulf, King Arthur, the Knights of the Round Table, and Robin Hood, but the Greeks give us Haracles, Perseus, Theseus, Jason, Achilles, Agamemnon, Odysseus, etc. etc. etc. There's simply more there. And keeps... there were castles in Greece, just like there were castles in the Middle Ages. In fact, King Minos's castle in Greece was both a castle as well as a dungeon.
Now maybe you'll argue that it wasn't until the middle ages that plate armor was invented, that metal craft wasn't perfected until then, or that there are a few more monsters that were picked up through the vikings, or the Americans, or the Arabs, but realistically, the cultural influences from those sources is not felt as greatly as the Greek culture is in most fantasy literature (granted, there are exceptions, but I'm making a general statement here). In fact, in terms of cultural influences, I would further argue that no culture in the history of the world was more influential on our own culture than ancient Greece. It is for all of these reasons that I believe that ancient Greece itself is a long-overlooked source of fantasy fiction. Unsurprisingly, this is an avenue that I am currently exploring, and you'll see the fruits of my experimentation in the coming months.