Until the day comes that we have DNA technology (and theory) advanced to the point where we can look at the genetic lineage of large groups of people, really the best indicator we have for cultural decent is language.
Now language isn't perfect in this regard. For instance, there are a lot of people indigenous to the Americas whose language has been lost (or nearly so), and speak English or Spanish instead. There's also the Pygmies, who probably had a very unique language of their own originally, but today speak Niger-Congo derived languages (albeit with some intriguing holdovers). However, this in itself can be viewed as a good indicator of how thouroughly their culture got absorbed into the culture of the new languages.
So I think it is quite fair to view anybody speaking a modern language derived from ancient Greek as a cultural descendent of the ancient Greeks. It is also quite fair to view anybody speaking a Romance language as cultural descendents of the Romans. As Samuel Johnston said, Language is the pedigree of Nations