The answer is rather paradoxical........it is both Yes and No.
Modern Greece was born in 1821 with The Greek War of Independence-(1821, for Greeks, is like 1776 for Americans). The Greek nation-state was born around 1828 with its First Capital on Aegina Island-(about 30 miles South of Athens and just North of the Peloponnese). Initially, the Modern Greek nation-state was primarily comprised of the Southern Greek mainland, parts of the Central mainland, Athens and the Cyclades-(the archipelago next to Mainland Greece). It would take 90 years-(from the 1820's until the end of the First World War) for what is today, the Modern Greek nation-state, to come into existence, that is to say, the inclusion of the Northern mainland, Crete and many of the Aegean islands. There was a brief return of the Ionian city of Smyrna-(present-day Izmir, Turkey) between 1919-1922, which was loosely affiliated with the Modern Greek nation-state under The Treaty of Sevres. However, other historic Greek territorial centers, such as Constantinople-(i.e. The "Old City" or section of Istanbul, namely the European side of the city), the Northern Anatolian Black Sea coast, parts of the Turkish Aegean coast and especially, the island of Cyprus, never achieved reunification with the Modern Greek nation-state. Today, much of what has been geographically and historically noted, has led to the current Modern Greek nation-state.
There are 10 million ethnic Greeks who live in Greece proper and approximately 800,000 ethnic Greeks who live in the Southern half of the island of Cyprus (included in the 800,000 are percentages of Greek Cypriots who once lived on the Northern half of the island, though were displaced after the 1974 Turkish invasion of Cyprus). There are tens of thousands-(perhaps even more) of Greeks who live in Southern Albania-(bordering Northwest Greece) and approximately 4 million persons of Greek descent who live in various Diaspora lands, such as, the United States, Canada and Australia. This is the contemporary historical and demographic reality of the Modern Greek.
As for the genealogical or ancestral nature of the Modern Greek and whether or not Modern Greeks are anthropologically related to the Ancient Greeks, this has been an age old question and apparently, it is still up for debate.
One must remember that the country of Greece-(as well as centuries old Greek Diaspora lands), experienced waves of conquerors, Traders and settlers from Antiquity, to the present-day. Minoan Cretans, Illyrians/(Albanians), Persians, Egyptians, Romans, Phoenicians, Mongol-Huns, Jews, Armenians, Georgians, Arabs, Slavs, Romanian Vlachs, Frankish Crusaders, Germanic Goths, Venetian Italians, as well as Seljuk and Ottoman Mongol-Turks, have all left a partial imprint on Greek history and on the Greeks themselves. I personally would be very surprised if there was a single Greek on this planet who could prove that he or she has a 100% uninterrupted Hellenic genetic line dating to Mycenaean times-(or earlier). This, (due to the long and complex demographic history that was discussed), would be a near genetic impossibility. However, I would be equally surprised if genetic tests revealed that the majority of Greeks living on this earth had very little or no genetic connection with the Ancient Greeks; this, would also be a near genetic impossibility.
In the 3500 plus year history of the Greeks, there is absolutely NO evidence for the widespread disappearance of the Hellenic race. Even with the well documented settlements and conquests that the Greeks endured over the millennia, there is ZERO proof of such a widespread population disappearance. Those who make such unsupportable claims, are either very ignorant of Greek History, very anti-Hellenic, or a little of both.
There was, however, one major and provable exception. We know from history that many Greeks living in Asia Minor-(present-day Turkey), were forcibly assimilated into the Turkish state and culture-(namely through intermarriage, though by other means as well) during the Ottoman Empire. Many-(though certainly not all) Asia Minor Greeks were thoroughly Ottomanized with regard to language, religion and overall self-identification as the centuries progressed-(this was especially true for the Janissare Corps). The Seljuk and Ottoman Mongol-Turks deliberately depopulated much of historic Greek Asia Minor, though the majority of the Hellenic population living in Greece proper and Cyprus.... largely survived Universal Ottomanization. In other words, much of historic Greek Asia Minor did indeed disappear with The Ottoman Empire, though there is barely any legitimate historical or genealogical evidence which proves that Modern Greek peoples-(on a widespread basis) converted to Christianity from Islam or originally descended from other lands, such as Albania or Slavic Europe. Again, those who make such unfounded claims, are fundamentally ignorant of Greek History. It is one thing to say that the majority of Modern Greeks have varying percentages of mixed ethnic descent-(such as having Albanian, Mongol-Turkish, Slavic or Italian heritage), yet, it is quite another thing to say that the majority of Modern Greeks have absolutely no genealogical continuity and connection with their ancient past without providing any legitimate historical, anthropological or genetic proof.
Overall, it is plausible to suggest that Modern Greeks have some fraction of mixed ethnic heritage, however, it is equally plausible to suggest that the Modern Greek has a sizable-(but not necessarily total) connection with his or her distant ancient ancestry. The annals of History, combined with contemporary advancements in archaeological and genetic technologies, can provide us with a wealth of evidence when looking into and examining this age old complex question.