For the earliest years of the Islamic Empire, under the Umayyad dynasty, we don't have very much art surviving. The best of it is the elaborate mosaics on the Dome of the Rock mosque in Jerusalem and on the Great Mosque in Damascus. These mosaics are done in a Roman style, probably by Roman craftsmen.
Another mosaic from the Great Mosque in Damascus
(about 710 AD)
But already we can see one big difference between Roman art and Islamic art: the followers of Islam, like the Jews, took seriously the idea that you should not make graven images, and although these mosaics show plants and buildings they do not show people or animals.
By the Abbasid period, even plants and buildings were frowned on. Most of the art was geometric designs. A lot of these designs seem to be from fabric patterns. The Arabs, because they were nomadic, had always relied on carpets and hangings for decoration. Now that they lived in buildings, they used those same familiar patterns only in stone or tile. They often used calligraphy (beautiful writing) of verses from the Koran to decorate buildings, plates, and vases.
In this period, also, the focus of the Islamic Empire shifted from Damascus and the old Roman territory east to Baghdad and the old Sassanian territory. So the art also became more Persian and less Roman.
By about 1000 AD, the Islamic empire was breaking up into smaller states, and each state developed its own art style. There are individual styles for Spain, the Maghreb, Egypt, the Ottoman Empire, and Persia.
Mosque in Cordoba, Spain
In some of these places, the iconoclastic rules against using pictures of things or people were relaxed as time went on. In Persia (modern Iran), painters made beautiful little miniature paintings of people at court, and of famous people from history.
Iskander (Alexander the Great),
Persian miniature from Herat, 1400s AD
To find out more about Islamic art, check out these books from Amazon.com or from your library:
Hands-On Ancient People, Volume 1: Art Activities about Mesopotamia, Egypt, and Islam, by Yvonne Merrill and Mary Simpson. Art projects for kids, though the directions are really aimed at teachers or parents.