Classic Greece is sometimes called the "crucible of western civilization" due to its cultural legacy. There are many different achievements that one could look to as distinctly Greek in origin, but the two that come to mind first are Democracy and Architecture.
If you've ever gazed up at a government building adorn with columns or enjoyed a great last row seat in a movie theater you have the Greeks to thank. The Greeks established some very fundemental architectural principals in their buildings and monuments. Adorning them with sculptures or great columns is the most recognzied, but the use of symmetry is something that the Greeks excelled in as well. The Greeks also created the first acoustically augmented outdoor theaters, a concept incorperated into every indoor and outdoor theater today.
Democracy is also an invention credited to the Greeks. Their form of Democracy was direct where the participants voted directly on legislative film. People who voted in these elections were cross-class, but always male. Only those who participated in military training could vote, but the idea of the common people taking control of their own government came from Greece.