Don't say that!
What I'm saying is that looking at Byzantium and saying "they're Greek!" or "they're Roman" can only have a meaning insofar as what Greek or Roman meant at the time you're talking about (and since Byzantium spawns a long period of history, say the Makedonian dynasty and the Palaiologos period are two very different things as well).
If your conclusion is that Byzantium is Greek, it doesn't suddenly mean that there is more continuity with Greek polities of an arbitrary point in time in the past than with Roman polities of that time, or vice versa. The whole Hellenic cultural sphere was a part of the unified Roman Empire for a long time and this resulted in strong Greek influences on Roman culture and strong Roman influences on Greek culture. Medieval Byzantine culture was also affected by Slavic, Syriac and Armenian influences in a non-negligible way.
The bottom line is that cultures aren't unchanging entities that move like threads through history while being "adopted" by certain states.