It might earnestly be easier to ask what of our modern society was not influenced by the ancient Greeks. The thriving culture of Greece developed over thousands of years, and many historians consider it to be the pioneer of modern Western culture.

Most importantly, the Greeks developed many of the governmental procedures and practices that we could scarcely imagine life without in this day and age. Democracy, jury trial, and equality in the eyes of the law can all trace their origins back to ancient Greek culture.

Ancient Greece was also a haven for intellectual thinking. The Greeks developed scientific pursuits such as mathematics, biology, and medicine. They also focused their intellect on the arts. In fact, much of Greek art and theater sets the standard for how we judge art today.

It is difficult to overstate the contributions the ancient Greeks gave to Western civilization. A few of their important contributions include the development of history, literature, and philosophy. The Greek historian Herodotus--most famous for his Histories--has been nicknamed the "Father of History" for his commitment to discerning the truth about the stories the Greeks believed about their history. Thucydides, another Greek historian, pioneered the interpretation of history as a series of political events. 

The Greek poet Homer birthed Western literature with his epics, the Iliad and the Odyssey. These works remain among the most significant in history and influenced later authors such as Virgil, Dante, Shakespeare, and Milton.

Finally, one cannot study philosophy without engaging with Greek philosophers such as Socrates, Plato, and Aristotle. Each of these men sought to find truth through moral, theoretical, and/or natural philosophy.

The contributions the Greeks gave Western civilization

The Greeks are rightly seen as a founding influence on western civilization. Athenian participatory democracy was a form of government the West long sought to emulate. The tradition of criticism of that same system and of society that first began with the philosophy of Socrates is another contribution. Greek philosophy has been very influential; the pre-Socratics and their attempts at naturalistic explanations laid the foundation for scientific inquiry. The philosophical idealism of Plato has been a powerful influence throughout Western history as have the writings of his student Aristotle. Aristotle's ideas about biology weren't eclipsed in some cases until the early 19th century.

Beyond this there is the importance of all of Greek mythology and literature, which became canonical in the Western tradition and echoed time and time again in countless works of fiction and non-fiction. The Greek alphabet became the framework for the English alphabet and Greek stories became foundational for the West more generally. Though it was adapted to Roman purposes, Greek mythology and thought was embedded in the Roman empire and this is one of the ways it first spread. Greek writings were also preserved in the Islamic tradition and it is these Arabic translations that helped spawn a revival of classical ideas during the Renaissance.

In essence, one can argue that Western civilization is founded on the ideas of the ancient Greeks.