Until the Age of Nationalism during which Greece gained its independence from the Ottomans, the people we consider Greeks today called themselves Romans, which makes since because the Byzantine Empire, centered around the Greek city of Constantinople, was the sole legitimate "Roman" state by 1453. The shift from "Romaioi" to "Hellenes" (i.e. Greeks) was the result of a reimagining of the national myth which viewed modern Greece as more of a descendant of Classical Greece centered around Athens rather than the Byzantine Empire centered around Constantinople, which 400 years fell so helplessly to their current enemies, the Turks. This romanticizing of pre-Roman Greece also helped to attract foreign volunteers in the war of independence, many of whom were Philhellenes with respect to the ancient Poleis and the Hellenistic kingdoms rather than the Byzantine Empire, which was often portrayed in a negative light in Western European historiography of the time. So back to your original question, essentially, the modern Greeks are the direct descendants of the Byzantine people.