the areas where most greeks lived was conquered by rome way before the byzantine period. indeed it was conquered well before the roman imperial period. even before the empire split into western and eastern halves the lingua franca of the empire was greek. after the split the great majority of the eastern empire was south of the jirecek line:

http://en.wikipedia.org/wiki/Jire%C4%8Dek_Line

greek speakers back then didn't seem them selves as part of a greek nation. they saw themselves as part of a nation of romans, using roman law but greek culturally. greek orthodox christianity was a big part of that identity. they certainly didn't consider themselves 'byzantine'. this term was invented by the historian gibbon in the 18th century to distinguish between the early roman empire and the later, eastern roman empire. greeks of byzantium called their home county 'romania' and themselves 'romaioi'(romans). greeks sometimes still refer to themselves as 'romaioi' even to this day.

i have an anecdote that might be illustrative. i should mention that i am of greek heritage and my family can speak greek. some years ago my sister was visiting istanbul on holiday and happen to run into a local greek (there are still a few thousand greeks left living there). she asked him, in greek, how it was to be a greek in istanbul (or something like that) and he replied that was wasn't a greek, he was a 'roman'(romaios). he called greeks who lived in greece itself greeks, but he considered himself something different to them. i think his attitude is partly motivated by a desire to be linked to the 'glory' of rome, a sort of imperial nationalism that still reverberates even after all these centuries have passed, even in the minds of many modern greeks.