Do non-European Cultures have a "Medieval" Period?

I have a daily Google alert for the word "medieval". Today it included a link to a Princeton University Symposium on Medieval Japanese art. That got me thinking: is it really correct to associate the word medieval--which describes the time between classical European history and the Renaissance (roughly)--to a concurrent time period in a non-European culture or society's history? I suppose it's more correct than alluding to the same period in Japanese history as the "Middle Ages", but then again, the term "medieval" is essentially a synonym for "middle ages." So how has it come to pass that we refer to a Japanese Medieval period which may or not have truly been "in the middle" of anything? Is it a relic of Eurocentricity?