Apparently, simple belief in God has little positive or negative correlation with economic growth; belief in heaven correlates positively, and belief in hell correlates with even bigger economic growth. The effect is apparently strongest among developing countries.
Researchers are not quite sure why this is. One speculation is that fear of punishment is a more primal motivation in human beings, and thus a fear of eternal damnation has a strong power to motivate people. So if one believes that one will be punished in the hereafter if one breaks the rules of morality, one tends to break them less often. The connection to economic growth might then consist in the fact that people who break such rules less often tend to cooperate more, work harder, cheat less, and so on--all factors in encouraging economic growth.
I find this plausible, though of course the issue is quite complicated: there are many factors influencing economic growth. Even if a link between religion and capitalism is conceded, however, it would still leave wide open the question of what to do about it. How large-scale shifts in belief or in culture take place is still largely unknown. How one might try to affect such shifts is thus even less well understood.
As an aside, I note that in my Actual Ethics, I made a related claim:
Good judgment develops [...] not only by enjoying the freedom to exercise it, but also by being required to take responsibility for its exercise. [...] Another way of making the same point: if you were going to create your own new religion, one requiring people to sacrifice and change their otherwise everyday behavior, it would help to have a hell. Promises of good things to come if one behaves the way your religion prescribes will take you some distance, more with some people and less with others; but your efforts will be considerably aided if you also have punishment for bad behavior. (pp. 11-12)Not quite the same claim, and I was talking about the development of human judgment, not economic growth. But both seem to agree that (1) human beings respond to incentives and (2) negative incentives for unwanted behavior are at least as important, if not more, than positive incentives for wanted behavior.
UPDATE 11/27/09: This post generated a bit of discussion at Kruse Kronicle. I find myself siding with Michael Kruse in the exchange. I would add to that discussion only the recommendation of a book I found quite interesting on the subject: The Bottomless Well by Peter Huber and Mark P. Mills.