Romans had a begrudging respect for the Greek states so they did not destroy them neither their culture.
Step-by-step explanation:
The Greeks were a major point of influence for the Romans in terms of culture and architecture and much of it was directly taken from the Greek city states in fact.
By the time they were able to conquer the Greek states they were not as powerful as they used to be but they still had respect for the cities.
They did attempt to conserve their culture and brought old Greek books back to their cities to translate and work on them.
They also let the cultural activities in Greece flourish under the roman rule.