The influences of the ancient Greeks upon Eastern Asia and other parts of the world is more indirect than in Europe. In Europe the Romans became very interested in Greek culture and ideas and incorporated them, then the Romans conquered territory and spread those ideas. With regard to Asia, Alexander the Great did spread Greek influence and culture in his conquests, but his empire was so short lived and then spit into three parts. The Macedonian conquest also did not try to impose a new social or cultural order on those conquered.

But this was the start of the Hellenistic period. Greek trade spread Greek ideas and culture throughout the Middle East, Northern Africa (largely Egypt) and even into India. Anceint greek coins have been found in both regions. Greek language became much simpler as it spread an more people began speaking it as a common tongue. This version of Greek is called koine. It became the language of educated people in the Eastern Roman Empire. The New Testament is written in this dialect. Greek trade in the Middle East arguably helped pave the way for the silk roads and greater interaction between East and West. It's advances in mathematics were stepping stones for greater advances in order areas.

In short, I suppose one could say that in Western society Greek ideas are in some respects the foundation, the starting point of how we understand thing. People always look to Greece for understanding democracy in the West. But in the East it did not replace the foundation but was a useful stepping stone and other people bounced ideas off the Greeks and exchanged wares as well.

If you need more specifics I can dig for them.