The Greeks had been establishing colonies in Sicily and the Italian mainland since the 8th century BC. Rome conquered Sicily in 264 BC and the Greek mainland in 146 BC. I would imagine the exposure to Greek culture was soon complete:
Wiki: "The Greek peninsula came under Roman rule in 146 BC, Macedonia being a Roman province, while southern Greece came under the surveillance of Macedonia’s praefect. However, some Greek poleis managed to maintain a partial independence and avoid taxation. The Aegean islands were added to this territory in 133 BC. Athens and other Greek cities revolted in 88 BC, and the peninsula was crushed by the Roman general Sulla. The Roman civil wars devastated the land even further, until Augustus organized the peninsula as the province of Achaea in 27 BC.
Greece was the key eastern province of the Roman Empire, as the Roman culture had long been in fact Greco-Roman. The Greek language served as a lingua franca in the East and in Italy, and many Greek intellectuals such as Galen would perform most of their work in Rome.
Several emperors contributed new buildings to Greek cities, especially in the Athenian agora, where the Agrippeia of Marcus Vipsanius Agrippa, the Library of Titus Flavius Pantaenus, and the Tower of the Winds, among others, were built. Life in Greece continued under the Roman Empire much the same as it had previously. Roman culture was highly influenced by the Greeks; as Horace said, Graecia capta ferum victorem cepit. (Translation: Captive Greece captured her rude conqueror.) The epics of Homer inspired the Aeneid of Virgil, and authors such as Seneca the younger wrote using Greek styles. The Roman nobles who regarded the Greeks as backwards and petty, were the main political opponents of Roman heroes such as Scipio Africanus, who tended to study philosophy and regard Greek culture and science as an example to be followed. Similarly, most Roman emperors maintained an admiration for things Greek in nature. The Roman Emperor Nero visited Greece in 66 AD, and performed at the Ancient Olympic Games, despite the rules against non-Greek participation. He was, of course, honoured with a victory in every contest, and in 67 AD he proclaimed the freedom of the Greeks at the Isthmian Games in Corinth, just as Flamininus had over 200 years previously. Hadrian was also particularly fond of the Greeks; before he became emperor he served as an eponymous archon of Athens. He also built his Arch of Hadrian there."