As people have said, none of them were descended from Constantine. Legitimacy mattered but it was more a question of having been born to the imperial circles/family than it was to have belonged to any particular family.
http://en.wikipedia.org/wiki/Born_in_the_purple
For example, Michael IV (1041-42) was a relatively obscure relation of a powerful eunuch and the former emperor (who himself had been a relation of the powerful eunuch John, long story) and he married the "born-in-the-purple" Macedonian princess/empress Theodora but after gaining the imperial position attempted to depose both Theodora and her sister Zoe. Even though he was emperor, the people and the aristocracy did not tolerate his abuse of the
legitimate princesses.
Likewise, John VI Kantacuzes (1347-1354) won a civil war to become emperor but made his young rival co-emperor instead of killing him because of the importance of legitimacy.
That being said, it always had to do with power and if someone
could they certainly would have seized or taken power. And frequently a general would approach the imperial city and scheme with some elements of the church or the military to become emperor, easy as that.
The Macedonians are interesting in that they seized power entirely through intrigue and assassination but then engaged in a concentrated propaganda campaign to proclaim their dynasty as always having been chosen to lead God's people through the dignity of the imperial office (one example of this is a biography of Basil I that describes eagles shielding him from the harsh Mediterranean sun during his youth). And after they had seized power, they de facto and de jure became legitimate and after a few generations it was very well established. Byzantium never had a dynasty rule more than 100 years and only if we include adoptions. Longest ruling dynasty were the Palaiologans. Constantine's own dynasty only ruled until 363 , with Valentine I seizing power in 364 and beginning the Valentinian dynasty.
Byzantium wasn't a feudal state, though it did have feudal elements in the late period. So the question of dynasty , etc, or being a blood descendent of a former ruler wasn't that important. Until the late period of disintegration, the administration was highly centralized and conceptualized around an idea of a state versus an idea of a dynasty or family. I can't remember the name in question but there was a big controversy when the Western wife of one emperor tried to divide up the empire between her children a la gavelkind when the emperor died, the commentators thought she was crazy for thinking that the empire was the personal property of any one person or family. That was the case, the emperor did not own the land, he managed the land and if he did it badly, then he was a tyrant, in theory. That is why panegyrics focused on the emperors' charity or benevolence or piety or philantropia. The most legitimate dynasty was the most successful and philanthropic dynasty.
