Previous studies on ultrasound-propelled nano- and microparticles have considered only systems in which the particle orientation is perpendicular to the direction of propagation of the ultrasound. However, in future applications of these particles, they will typically be able to attain other orientations. Therefore, using direct acoustofluidic simulations, here we study how the propulsion of triangular nano- and microparticles, which are known to have a particularly efficient acoustic propulsion and are therefore promising candidates for future applications, depends on their orientation relative to the propagation direction of a traveling ultrasound wave. Our results reveal that the propulsion of the particles depends strongly on their orientation relative to the direction of wave propagation and that the particles tend to orient perpendicularly to the wave direction. We also address the orientation-averaged translational and angular velocities of the particles, which correspond to the particles' effective propulsion for an isotropic exposure to ultrasound. Our results allow assessment of how free ultrasound-propelled colloidal particles move in three spatial dimensions and thus constitute an important step toward the realization of envisaged future applications of such particles.