A technique based on the combination of Fourier pseudospectral method and the perfectly matched layer (PML) is developed to simulate transient acoustic wave propagation in multidimensional, inhomogeneous, absorptive media. Instead of the finite difference approximation in the conventional finite-difference time-domain (FDTD) method, this technique uses trigonometric functions, through an FFT (fast Fourier transform) algorithm, to represent the spatial derivatives in partial differential equations. Traditionally the Fourier pseudospectral method is used only for spatially periodic problems because the use of FFT implies periodicity. In order to overcome this limitation, the perfectly matched layer is used to attenuate the waves from other periods, thus allowing the method to be applicable to unbounded media. This new algorithm, referred to as the pseudospectral time-domain (PSTD) algorithm, is developed to solve large-scale problems for acoustic waves. It has an infinite order of accuracy in the spatial derivatives, and thus requires much fewer unknowns than the conventional FDTD method. Numerical results confirms the efficacy of the PSTD method.