Turning to religion, Mr Beattie asks: “Why don’t Islamic countries get rich?” Ah, he replies, some of them do. Islam is often held up as inimical to economic progress. That is nonsense, he says. The Muslim Hausa have provided some of Nigeria’s most successful traders for centuries. “Had Max Weber lived among the Hausa”, Mr Beattie sniffs, “he might well have concluded that Muslims were good for growth and constructed his convoluted psychological theories around the tenets of Islam.” The author picks his way through religious texts, history and modern commercial practice to argue that there is no reason to draw a firm causal connection between any faith and economic progress.
Here is another review for False Economy, from Business Week, that discuss the Islamic question:
Beattie, who studied history at Oxford University and economics at Cambridge, draws on both disciplines to overturn assumptions about the evolution of the global economy. For example, the data do not support the belief that Islamic societies inherently perform worse than other nations, or for that matter that there is any correlation between religion and growth. Malaysia has both a strong Islamic identity and a modern economy. Religion is an obstacle only when development is blocked in God's name, often in self-defense by those who hold power, Beattie argues.
In looking up information about this book, I came across an old blog post at Aqoul that discusses a Financial Times article Beattie wrote on the same subject, most likely becoming part of his research for the book now published. (Unfortunately, the FT article is no longer available on the FT website.) However, the Aqoul post refers to the original study Beattie was writing about, Religion, Culture and Economic Performance by Marcus Noland at the Institute for International Economics, which was published in August 2003. In that study, Noland found that:
The results with respect to Islam do not support the notion that it is inimical to growth. On the contrary, virtually every statistically significant coefficient on Muslim population shares reported in this paper—in both cross-country and within-country statistical analyses—is positive. If anything, Islam promotes growth.
(A similar paper by Noland and Howard Pack, Islam, Globalization, and Economic Performance in the Middle East (published June 2004), came to the same conclusion.)
So, the partial answer to the question, "Why don't Islamic countries get rich?" is, "It's not Islam's fault." To answer the question more thoroughly requires a more conventional economic analysis. (I hadn't originally planned a part two, but I feel one may be necessary at this point.)
Update (8 May 2011): I think the events of recent months (the "Arab Spring" and especially the examples of Tunisia and Egypt) have shown that the problem with respect to economic growth in Muslim countries is not Islam itself, but the authoritarian control by governments over economic activity, especially when that control is used to stifle the economic activity of the average Muslim in favor of cronyism. Don't forget that the Tunisian revolution, which started the Arab Spring, started when a young man, Mohamed Bouazizi, set himself on fire because local officials wouldn't allow him to support himself and his family by selling fruit and vegetables from a pushcart.