What aspects of Greek and Etruscan cultures can be seen in ancient Rome?

Because the Etruscans were influenced by the Greeks, it can be difficult to separate out the Greek and Etruscan influences on ancient Rome. However, there are certain distinct areas in which Etruscan influence can be seen. First, although we tend to think of gladiatorial displays as Roman, they actually were originally an Etruscan practice. Next, much of the engineering and technological prowess we associate with the Roman Empire had its origins in Etruscan practices. The Etruscans pioneered the development of road and bridge networks, use of vaults and arches in architecture, and underground sewer systems. 
The Greek influence on Roman culture was equally pervasive. This was famously expressed by Horace in the line:

Conquered Greece took captive her savage conqueror
and brought her arts into rustic Latium. (II.i: 156-157)

The Roman gods were, to a large degree, assimilated to the Greek pantheon and mythology. Roman students studied the Greek language and learned Greek literature, rhetoric, and philosophy in schools. Roman theater originally consisted mainly of adaptations and translations of Greek plays and similarly Roman art, especially sculpture, imitated Greek models.