The foundation of many things in the Western world come from Ancient Greek civilization. Many philosophies, sports, arts and even medical treatments have their basis in Ancient Greek traditions. Contributions to our current civilization still continue to exist today.
Perhaps the biggest influence Ancient Greece has had in the current world is that of government. The first known democracy was created in Greece and gave political power to the citizens. This culture has continued, and the power to elect continues to be held by the people.
Philosophies first created in Ancient Greece are also still heavily discussed and debated, especially those by Aristotle, Plato and Socrates. Their philosophies and theories have helped to shape generation after generation, and will likely continue to do so.
Hippocrates (from where the Hippocratic Oath stems) is often referred to as the father of medicine and many medical treatments have their origins in Ancient Greece. Of course, procedures now have been refined, but their basis still lies in ancient times.
Sports also played a big part in Ancient Greek life, and Greece hosted the very first Olympic Games, which of course are still held today every four years. Many of the sports found in the modern day Olympics either closely resemble or are the same as those competed in at the first Olympics.
You can look at almost every part of life and find a connection, an influence from Ancient Greek times. Despite lacking the technology we currently have, they were an organized and hard working society, and many of their notions were sound, explaining why so many of their traditions have been incorporated into the current world.
Although long gone, Ancient Greece continues to live on in our daily lives, traditions melding seamlessly with the current world, creating a strong foundation for us to build upon.