One period from the history of Western civilization that resembles the Greek "Dark Age" is the early medieval period in Western Europe. This is a period that lasted roughly from the "fall" of the Roman Empire in 476 until 1000 AD.
In both times and places, there was a drop in economic activity and population. Both of these drops occurred after one dominant civilization had fallen. With the fall of the dominant civilization, there was less security for people, which meant that it was harder to make a living and harder to trade. As economic conditions worsened, it was generally harder to keep population numbers up as well.
Thus, the early Middle Ages in Western Europe can be compared to the Greek Dark Age because both were times of economic and social hardship that followed the collapse of major powers in the respective areas.