The ancient Macedonians were completely destroyed and slaughtered by the Romans and the few that survived left Macedonia for mountainous regions or the Balkans. Most of the large macedonian cities were abandoned and destroyed. Nobody can say today he is a pure descendant of them. Macedonia was then inhabited by slavic groups of people, basically Bulgarians and Serbs or other south Slavs. The Macedonians of today are a mixture of Bulgarians (mostly) and Serbs and maybe some few descendants of ancient Macedonians.
The problem of Greece is that it has put in Macedonia the majority of the christian Turks, Persians, Arabs, Armenians etc that they brought from Asia Minor at 1923 and they want to make them believe they are descendants of Alexander the Great and so that they are related to ancient Greece. Karamanlis, the prime minister that put veto on the entry of Republic of Macedonia in NATO, is one of them and comes from the turkish-azeri tribe of the Karamans. Some others from the same tribe (also with the name Karamanli) were the governors of Libya at the 18th century. You can also find people in Turkey with that name. These people were never greek and they didn’t even speak greek before moving to Greece.
In my opinion, Macedonia must accept its slavic identity and try to inform the slavic people that still live in greek part of Macedonia, instead of saying they are the descendants of Alexander the Great. There are still lots of people in greek part of Macedonia who have slavic origin and have no idea about it. They try to convince them that they are all the same with the Turks and rest anatolians that came from Asia Minor at 1923. They changed the slavic names of villages, slavic surnames etc and they obliged them to speak only greek. But Macedonia has not done much to help these people wake up. And also the other slavic countries that have minorities here have not done much to help them.