I am having difficulty with this one too. No matter what I do, I always seem to lose 3 cities.
I have tried different difficulties between Warlord and Emperor in case the number of cities might increase with difficulty (it does not seem to matter).
My approach is to take my initial continent in the first 2-3 eras (depending on size and number of opponents) and then expand at the end of an era planting several settlers on other continents close to AI players so that they have low loyalty. In my latest attempt, I planted 7 cities within the 5 turns before the end of an era. As the Dark age pops, 3 cities flip, and then as my turn starts, 3 more cities flip (total of 6) with the 7th ready to flip 1 turn later...and nothing, no achievement.
It seems that the actual Dark Age game mechanic must cause 4 cities to flip, but I am unsure how to make this occur. Can someone who has completed this help with some advice?