Surface-acoustic-wave (SAW) devices form an important class of acoustofluidic devices, in which acoustic waves are generated and propagate along the surface of a piezoelectric substrate. Despite their widespread use, only a few fully three-dimensional (3D) numerical simulations have been reported in the literature. In this paper, we present a 3D numerical simulation taking into account the electromechanical fields of the piezoelectric SAW device, the acoustic displacement field in the attached elastic material, in which a liquid-filled microchannel is embedded, the acoustic fields inside the microchannel, and the resulting acoustic radiation force and streaming-induced drag force acting on micro- and nanoparticles suspended in the microchannel. A specific device design is presented, for which numerical predictions of the acoustic resonances and the acoustophoretic response of suspended microparticles in three dimensions are successfully compared with experimental observations. The simulations provide a physical explanation of the observed qualitative difference between devices with acoustically soft and hard lids in terms of traveling and standing waves, respectively. The simulations also correctly predict the existence and position of the observed in-plane streaming-flow rolls. The simulation model presented may be useful in the development of SAW devices optimized for various acoustofluidic tasks.