Well, Greek democracy isn't really where modern democracy came from--there was a significant interruption, and modern democracy is from a separate tradition and quite different ideologically (human rights vs citizen egalitarianism). The connection to Greece is in many respects rhetorical--"Greece" has been, in literature and philosophy, the archetype for "democracy" for a very long time and so it was often held up by those advocating for or against democracy.

That being said, democratic forms of governing are very common. Hunter gatherers and other what we might call "low complexity" political societies often are democratic in decision making.

what you're referring to as Greek democracy was Athenian democracy (Corinth had democracy too, Sparta did not) and it doesn't really resemble modern democracy at all as understood by political theory. Its governing body was the Athenian Assembly and it was what is now called direct democracy – essentially government by referendum, the Assembly (composed exclusively of freeborn Greek males) would vote on pretty much every decision that concerned the city of Athens, contrasting with the modern democracy we're familiar with wherein citizens vote presidents/prime ministers/parliament/congress/senate etc. into public offices where they will make most of our decisions for us rather than make us vote on every piece of proposed legislation or what have you. A few countries such as Switzerland are some form of direct democracy today, mostly on a kind of federal level, but clearly it would be detrimental in most cases (remember classical Athens had a population of only 100,000 with only 30,000 approx. being adult Greek males who were eligible for voting in the Assembly).

It should be noted that democracy in Athens was intermittent; the "Golden Age" was under the leadership of Pericles, during which the Parthenon was built; after the Peloponnesian War an oligarchy known as the Thirty Tyrants seized power, the were subsequently overthrown and democracy returned.

The ancient Indian republics. Much before the Greeks.