The 14th century pandemic of bubonic plague known as the "Black Death" had a devastating effect on most of Eurasia, the Islamic world included. I hesitate to speculate about why the impact outside of Europe isn't as well known, other than the usual sorts of ways in which the pandemic was worked into the narration of European exceptionalism as one of the precursors that allowed Europe to overtake the rest of the world. I will say that while the pandemic was an economic and demographic disaster in the Islamic World, it did not entail the amount of social and cultural upheaval that it did in Europe.

The short answer to the question posed in your last sentence is that the pandemic appears to have been about as virulent and deadly in North Africa and Southwest Asia as it was in Europe. The Egyptian medieval chronicler al-Maqrīzī said that as many as 20,000 people died each day in Cairo from the disease in the winter of 1348-9; Michael Dols has estimated that this accounted for between 1/3 and 2/5 of the city's population (Dols, "General mortality," below). Stuart Borsch (whose monograph The Black Death in Egypt and England might be of interest) estimated that in the Middle East between 1/3 and 1/2 of the population perished.

As in Europe, the massive depopulation had far reaching effects in the Islamic world--historians have really only begun trying to fit these into the larger picture in the last couple of decades (again, this is probably one of the reasons you don't know much about this). In areas that were heavily dependent on irrigation, such as Egypt and Mesopotamia, the depopulation resulted in canals drying up and desertification (because no one was able to keep the canals maintained). Unlike in Europe, where serfs moved onto their deceased owners' lands, in places where tax farming prevailed it led to land abandonment.

(Tax farming is a system wherein nobles bid for the right to collect taxes from certain administrative regions; everything they're able to collect above that amount is their own profit. Since the tax collector's ability to bid again is based on their ability to procure an amount of tax they themselves have guaranteed, it reduces corruption on the administrative end; however, since the purpose is to raise that amount and a little extra, it was also tied to coercion and mistreatment of the peasantry who actually grew the crops that were to be sold and--since most of the workers weren't landowners--in times of poor production there was literally nothing keeping them from just walking off the estate.)

It's estimated by Borsch that it took Egypt over a century to regain the population and level of agricultural production that it had prior the Black Death; by this time the Ottomans--who were able to take advantage of the situation by restoring order in places badly affected by the pandemic--were knocking on Egypt's door ready to conquer in 1517.

One thing that did not happen as commonly in the Middle East as in Europe was the abandonment of cities for rural areas; another thing that seems to have been relatively infrequent was blaming outside groups (in Europe the Roma and Jews were usual targets) for having brought God's wrath.

There were no successful medical methods available to treat the plague until the era of antibiotics. European observers long derided Muslims as being "fatalistic" in the face of epidemic diseases, resorting to prayer and ascribing life or death to the will of God--I'm not sure this was much different in the Islamic World than it was in Europe during the Black Death, though. There is a long (and, unless you're really into theological debates, not very interesting) corpus of debates by Muslim theologians about whether contagion existed (very short version: the question was whether anyone but God could cause someone to get sick).

There was a strongly persistent belief that disease was thought to travel through the air via strong odors, interestingly one of the folk remedies employed was to either plant strong smelling flowers or to bring livestock indoors as the odors generated were believed to counter the mal aria (bad air). There is an entire section in Dols's book The Black Death in the Middle East about various talismans and magics that were employed--writing verses from the Qur'an on a slip of paper and either eating the paper or dipping it in oil and using the oil to anoint the foreheads of people; placing certain verses on the door in a manner similar to the Jewish mezuzot, and a host of other things. For my taste, a lot of scholars tend to treat these very dismissively--obviously, from a scientific perspective none of these methods have any validity, but neither did anything else. People tried everything they could to ward off the disease, because it was well known that once someone had contracted it, death was almost certain.

References and More Stuff

I highly recommend Borsch's book as an overview: Stuart J. Borsch, The Black Death in Egypt and England. A comparative economic analysis, Austin: University of Texas Press, 2005

You can also check out his article "The Black Death and the Human Impact on the Environment," which looks at the Buhayra province of Egypt here.

Michael Dols's The Black Death in the Middle East, Princeton University Press, 1977, is a bit dated but still a standard work.

See also his "The general mortality of the Black Death in the Mamluk Empire," in A. L. Udovitch (ed.), The Islamic Middle East, 700–1900, Princeton 1981, which I referenced above.