My comment is based on knowledge i have from being taught ancient history at undergrad by a native Greek.

It is hard to say that any "Greek" culture that exists today is actually directly related to the ancient greeks. This is due to the romanisation and overall cultural syncretism that occurred during the time of the empire. First, while based in Rome, greek cultual influence changed the dynamics of roman society and religion, and caused a rise in the bilingualism of both educated Greeks and Romans. Over time, more and more native Greeks began to move their emphasis on identity away from the polis and instead towards Rome, as they became citizens.

With the move of the capital to constatinople under constatine, the Greeks began to see themselves as truly roman. As many already viewed themselves as roman citizens and were culturally roman, the placing of the centre of roman identity within traditionally greek geography broke the last boundary needed for the greeks to become roman. This was compounded by the fall of the Western Roman Empire which meant that Rome itself became a foreign and un-roman place. It is therefore argued by my lecturer that up until the late renaissance, the people within the traditional greek sphere of the aegean would identify solely as roman, even after the take over of the ottomans. The difference in religion made these divisions clear. The ancient greeks and their culture was disgustingly pagan, the ottoman turks and occupiers were muslim, and the native roman population was orthodox christian.

Due to the fact many of these occupied people were never educated, the knowledge about the geographical and cultural history was stunted, which allowed interested parties from the West to begin defining their history for them. Much of the revolutionary feeling basely stemmed from the wish to rid themselves of the ottoman yolk, but was twisted by the educated westerners into an idealised "Greek" nationalism which stemmed from an anachronistic and foreign outlook. In the wars that followed, after the failure of the "Greeks" to recapture constantinople and free their brothers on the turkish side of the aegean, one of the agreements of peace was that the native "greeks" would be relocated to the now free Greek state. At this stage it is important to remember that the majority of the greek state were uneducated romans who cared nothing for ancient greece, but had their policy defined by the leaders and semi educated, who were heavily influenced by western idealism, and needed to foster a united national identity in order to be able to robustly fight the turks both in ideals and physically. This was done by implementing an education system which was based on the reinforcement of this foreign idea of greek nationalism, but was widely accepted by the uneducated masses, including those romans brought over from turkey.

So in reality the ancient greek culture fused and changed into roman culture during the roman empire, and died out as its own entity before the fall of the eastern roman empire. What we have today is based around an educated western traditional of idealised ancient greek culture, which in the area around the aegean is manifest in the Greek state but has been further affected by more modern politics and wars.