This paper presents an efficient method of approximating unsteady flows using a block-wise discrete spatial Fourier series for the modeling of three-dimensional non-axisymmetric flows without making any hypothesis about its temporal periodicity. The method aims at capturing the long-wavelength flow patterns that are present in many unsteady problems of industrial interest, such as compressor stability, with a drastic reduction in computational resources. The method is intended to be used to compute flows exhibiting large-scale instabilities and where the fundamental frequency of the problem is not known beforehand. The approach discretizes the domain using a finite number of blocks or passages, where the flow variables at the supposedly periodic boundaries are continuously updated using the spatial Fourier coefficients of a uniformly spaced set of reduced-passage domains. The NASA rotor 67 under the effect of distorted inflow conditions has been used as verification case to demonstrate the effectiveness and viability of the method. The comparison between the passage-spectral method and the full-annulus solution shows that sound solutions can be obtained with a low number of harmonics. The new method has also been applied to investigate the rotating stall inception of the NASA rotor 67 for distorted inlet flows near stall operating conditions. The method is shown to accurately reproduce the full-annulus solution with a few spatial harmonics, capturing the characteristic features of the complex flow induced by the tip leakage vortex breakdown. The computational cost in this application has been reduced by a factor of between three and seven. This number heavily depends on the ratio between the number of retained harmonics to the number of blades.