An extended Fourier pseudospectral time-domain (PSTD) method is presented to model atmospheric sound propagation by solving the linearized Euler equations. In this method, evaluation of spatial derivatives is based on an eigenfunction expansion. Evaluation on a spatial grid requires only two spatial points per wavelength. Time iteration is done using a low-storage optimized six-stage Runge-Kutta method. This method is applied to two-dimensional non-moving media models, one with screens and one for an urban canyon, with generally high accuracy in both amplitude and phase. For a moving atmosphere, accurate results have been obtained in models with both a uniform and a logarithmic wind velocity profile over a rigid ground surface and in the presence of a screen. The method has also been validated for three-dimensional sound propagation over a screen. For that application, the developed method is in the order of 100 times faster than the second-order-accurate FDTD solution to the linearized Euler equations. The method is found to be well suited for atmospheric sound propagation simulations where effects of complex meteorology and straight rigid boundary surfaces are to be investigated.