We present a novel accelerated Stokesian-dynamics algorithm for a system of spherical particles confined between two parallel walls. The algorithm is based on our recently developed Cartesian-representation formalism. A simplified asymptotic multipolar basis for Stokes flow in the wall-bounded domain is used to describe the far-field interactions between particles. Thus, only the near-field interactions between neighboring particles need to be exactly evaluated. By an appropriate transformation of the multipolar basis of Stokes flows, the problem is reduced to a sparse linear system of algebraic equations. The system is solved efficiently using a preconditioned conjugate-gradient method. Examples of numerical applications of our algorithm will be provided.