In this explanation, the Church was merely one player, and it wasn't the religious nature in itself which led to the lack of progress. Instead, it was the inward-looking nature of all of Western Europe which meant all players were more interested in gaining influence within that closed sphere rather than growth (Perhaps the reign of Charlemagne is an exception here).
There are numerous examples of this kind of behavior: the sack of Constantinople by the Crusaders in 1204 is probably outside the period of time usually classed as the "dark ages" but is still a good example of this behavior.
I'm pretty much in the Hawking/Dawkins camp regarding religion, I must admit.
Edit: Quoting Hawking, from the Wikipedia article on the conflict thesis: "There is a fundamental difference between religion, which is based on authority, [and] science, which is based on observation and reason. Science will win because it works." That was obvious to me when I was 12.
Except if you look at actual history of science and religion, it's way far from being that simple. Religions change, sometimes drastically, and are completely capable of overthrowing and even completely abolishing (see move from Catholicism to Protestantism) the authority structure. And science has plenty of authority issues, to the point where researches note that in some areas significant progress is made only when "old school" dies out and is replaced by "new school" - not because of reason, but because of physical limits of human existence. And some areas of science are very much influenced by political and other issues that have little to do with observation and reason.
So in practice both work in ways much more complicated than that.