What form of chess was played in seventh-century Arabia?
Wikipedia has an entry on the History of Chess. It states that the differences between what is now modern chess began in Europe in the 15th century. Besides the name changes for the pieces, the rules differed. The section covering The Indo-Arabic game appears to reference, Murray H.J.R. 1913. The history of chess. Oxford. reprint ISBN 0-936317-01-9

In early chess the moves of the pieces were:
Shāh: as King now.
Firzān: one square diagonally at a time (contrast modern Q).
Fīl: two squares diagonally (no more or less), but could jump over a piece in between (contrast modern B). This move meant that an individual fīl could move to only eight squares on the board, and no fīl could ever attack another fīl.
(Horseman): as Knight now.
Rukh: as Rook now.
Pawn: one square forwards (not two), capturing one square diagonally forward; promoted to firzān only.
The game could be won by 1. Mating the king; 2. By giving stalemate; 3. By capturing all the opponent's pieces. This kept down the number of draws.

The Murray book in its entirety is out of copyright, I found this section regarding Islam in India. He notes that the prophet supposedly denounced the game, and the difference in rules covered above on the Wikipedia page. This other site notes that Murray objects to the belief that Mohammed would have seen chess played in his lifetime, since the game is of Indian/Persian origin, and wouldn't have spread to the Islamic world until the fall of Persia.

&quot;most probably the Prophet had never heard of the existence of chess, since the Muhammadan jurists have been unable to settle the question of the legality of chess-playing by any direct decision of Muhammad as recorded in the Qur'an, or in authentic tradition. (HJR Murray 187.)&quot; It is difficult to state with any great certainty that chess was played in Arabia during the Prophet's lifetime; chess was after all an Indo-Iranian invention, and it is not unreasonable to assume that it entered Arabian culture through the conquest of Egypt and Western Asia after the Prophet's death. However, it is clear that chess had gained tremendous popularity amongst all classes and social spheres by the time of the Abbasid Caliphate, which took power in the year 750, still very early in Islamic history. It was at this time that religious scholars began to question the legality of playing chess.