I think I'll take a stab at making a list, this is just my opinion though so feel free to tear it apart and disagree vehemently.
I'll start with Achaemenid Persia, which while not able to conquer Greece remain the most powerful civilization until Alexander's invasion. So:
Persian Empire (559 BCE - about 333 BCE)
Greece - in the loosest term, including all the successor states of Alexander - was the dominant power until well after the founding of the Roman Republic, but was eventually eclipsed by the Romans. But I will use for my date the death of Pyrrhus of Epirus, so something like:
Greece (333 BCE - 272 BCE)
Rome, in my opinion was perhaps the greatest civilization in history, when taking into account all aspects of it, including its longevity compared to other empires. Rome was probably the greatest military power until the division of the empire after Theodosius' death, so:
Rome (272 BCE - 395 CE)
After Rome, the natural successor would have to be the Byzantine Empire, but while they survived until almost the end of the 15th century, I think there were more dominant powers long before Constantinople fell, such as the Islamic Caliphate. However I am going to (somewhat arbitrarily) set the date to the coronation of Charlemagne, so:
Byzantium/Constantinople (395 CE - 800 CE)
Even though it wasn't really a uniform civilization, I would put the Franks as the next great military power. I would consider them the most powerful group until Genghis Khan's Mongols, so:
The Franks (800 CE - 1206 CE)
While the Mongols and Franks didn't really come into much conflict, I think the Mongol Empire was an obviously more powerful civilization than the Franks during the 13th century. The Mongol Empire was huge and quite powerful, but it didn't really last very long. Almost immediately after Genghis Khan it began dissolving, so while there would be some overlap, I will say that:
Mongol Empire (1206 CE - 1282 CE)
This next one is mostly based on my personal opinion in the earlier years of its dominance, but I would say the next great power was the Holy Roman Empire, but more specifically the Habsburgs, so:
Holy Roman Empire (1282 CE - 1521 CE)
At this point I believe Spain became the dominant power, while not really a completely different group than the Habsburgs of Austria, I put the rise of Spain at the division of the Habsburg possesion in 1521, and consider them to be dominant until the rise of the British Empire, or possibly until the death of Charles II but to make the dates match I'll go with the first so:
Spain (1521 CE - 1707 CE)
The British Empire could arguably be the most powerful civilization from this point until World War I, and while I think that during the reign of Napoleon, France was the more dominant military, a single emperor doesn't put France ahead of Britain, so:
British Empire (1707 CE - 1918 CE)
From that point I will give credit to the United States, so:
United States of America (1918 CE - Present)
Whether the USA is currently as dominant as they have been in the past is not really a historical issue yet so I won't make any comments about that.
So here is my overall list:
Persian Empire (559 BCE - 333 BCE)
Greece (333 BCE - 272 BCE)
Rome (272 BCE - 395 CE)
Byzantine Empire (395 CE - 800 CE)
The Franks (800 CE - 1206 CE)
Mongol Empire (1206 CE - 1282 CE)
Holy Roman Empire (1282 CE - 1521 CE)
Spain (1521 CE - 1707 CE)
British Empire (1707 CE - 1918 CE)
United States of America (1918 CE - Present)