One of the most popular Renaissance instruments, the lute, has its roots in the middle eastern oud (or ud), which was probably made its way to Europe with the Moors when they conquered Spain in the 8th century AD. In addition to instruments such as these, and the bombardes, shawms, etc., the methods of playing them and thereby the music associated with them would certainly have made the journey as well.After the Crusades, many things that filtered back from the east became absorbed into fashionable Western European society. There were the obvious exotica and luxuries like spices, silks, perfumes, etc., but also the influences of their music and culture. There is scant European music available from the Middle Ages; one will quickly become familiar with the existing estampie, salterelli, & istanpitta if you start listening for them. The Dufay Collective & the New York Ensemble for Early Music are just two groups that have made several recordings of dance music of this period in the last few years.
In truth, I can recommend a number of recordings if you're interested; the common musical "feel" tends to be very eastern, and while this is based on the assumptions noted above, it does make sense within the context of the popular material culture of the time. A glance at manuscript illustrations of the 13th & 14th centuries shows us the fashion for turbaned head-dresses & bright, flowing silk robes; the most expensive (& therefore prestigious)delicacies & spices were imported from the east. Clearly this was a period in which sophistication often had a somewhat "exotic" flavour; in fact, most educated reconstructions of what mediaeval European food might have tasted like favour the opinion that it was a mix of sweet/spicy/salty tastes very like modern middle eastern foods!