In this paper, we present an efficient projection method to solve the three-dimensional time-dependent incompressible Navier-Stokes equations in primitive variables formulation using spectral approximations. This method is based on a modification of the algorithm proposed by Goda [J. Comp. Phys. 30 (1979) 76]. It brings an improvement by introducing a preliminary step for the pressure in order to allow a temporal evolution of the normal pressure gradient at the boundaries. Its efficiency is brought to the fore by comparison with the Goda's algorithm. The modified projection method is then applied to the simulation of complex three-dimensional flows in rotating cavities, involving either a throughflow or a differential rotation.