For one thing, Greek literature is the model for the literature of Western Civilization, that and Latin literature as well. Our ideas of heores and literary structure and mimetic qualities, our ideas of tragedy and the roots of comedy all arise from Greek precedents and the explanations of Aristotle.
This is an enormous topic. Here's what happened when I did a search of this topic in Google Books:
When trying to deal with a topic this large, it's often useful to try to find a bibliography (a listing of relevant publications). Even better is an annotated bibliography (in other words, a listing of relevant publications that tells you a little bit about each of them).
Here's what I found when I searched Google Books for "annotated bibliography ancient Greece":
Ancient Greek architecture also influenced and continues to influence construction practices and designs today. The inspiration for many of the public buildings in the United States comes from forms and structures created and first built by the Greeks.
The whole discipline of history comes to us from the Ancient Greeks. It was Herodotus and later Thucydides who came up with the idea of trying to explain why things happened rather than simply writing "history" in ways that were meant to either A) show the power of the gods or B) praise the rulers who allegedly made all good things happen.
Let us not forget too the huge debt we owe to the Ancient Greeks in terms of the massive scientific discoveries that were made during this time. So much of modern astronomy and mathematics is built upon discoveries that were made during the Ancient Greek civilisation.
Ancient Greece continues to influence Western Civilization. We have a justice system and governmental structure largely based on the Greeks. Even our economy is influenced by the ancient Greek monetary and trade methods. We also are still facinated with Greek mythology.