Cultural influence isn't just popular media. The language (Latin based), the government (democracy/senate), architecture (the white house), etc. They are the sum of a nation identity, how it appears, act, interact with others.
You can see Rome's traces in almost everything in the West, from entertainment to moral norms. Same thing in Asia, except they are Chinese, like the language (Japan, Korea, Singapore), architecture (Vietnam, Japan), entertainment (see that new Wukong game), etc.
The culture of a nation morphs and changes on the surface but deep down, it is based on a rather stable foundation. In the West, it is Greek-Roman, in the East, it is Chinese-Indian.