The upper class Romans were schooled in Greek letters. They were often tutored by educated Greek slaves, and finished their schooling in Greek cities like Athens and Rhodes.
Upper class Roman culture from the late Republic through the Empire does bear an influence of Hellenistic culture. A prime example is Virgil, whose Aeneid was a conscious imitation of Homer's Iliad and Odyssey.
That being said, Roman culture is not synonymous with Greek. Roman literature and philosophy may be influenced by the Greek, but it has its own tone (less abstract, more practical). Roman architecture may be based on Greek, but it has its own stories to tell (the use of arches, concrete).
Even some of the religious cults that were imported from the Hellenistic east were to some degree Romanized (the Cults of Isis and Bacchus had to find a Roman niche, for instance).
Then the majority of Romans, the Plebians, who were not educated in Greek ways, held to their native culture. Gladiator games and such would be their cultural experiences.