We present an efficient and accurate computational method for predicting three dimensional unsteady flows in multistage turbomachinery. Specifically, a time-linearized unsteady Navier-Stokes method is developed to solve the turbomachinery flutter problem, including unsteady blade row interaction. A time-linearized approach allows us to model each blade row with a computational grid spanning only a single blade passage. The aerodynamic interaction of blade rows is caused by propagation of acoustic, vortical and entropic waves in the working fluid. Each wave has a particular frequency and interblade phase angle related to the circumferential wavelength of the scattered wave and the frequency and the wavelength of the original wave that produced it. The waves propagating between the rows are modeled by exchanging information among various unsteady solutions at the interrow computational boundaries. Fortunately, a small number of unsteady waves and corresponding unsteady flow solutions must be retained in the model to compute accurately the unsteady aerodynamic response. This is important for the computational time, which is proportional to the number of unsteady solutions and the number of blade rows. In this paper, computational results are compared to those of a previously developed two-dimensional solver for an inviscid, compressible multistage compressor comprised of three blade rows. We also present the steady-state viscous solution and linearized unsteady solution for one and one half stages of a modern front stage compressor. Finally, we investigate the steady and multistage unsteady flow in the NASA Rotor 67 fan.