An efficient numerical scheme is developed to implement the inflow/outflow conditions for unsteady turbomachinery flows. The formulation gives exact non-reflective conditions which can be used with a time-harmonic code for calculating the unsteady flow about blades as well as the acoustic radiation. The method is applied to flows with a mean swirl. Upstream and downstream from the blade row, an asymptotic behavior of the scattered unsteady disturbances is obtained from an eigenmode analysis of the unsteady linearized Euler equations. The inflow/outflow conditions are developed as a filter which selects the physically acceptable modes and thus provides a non-reflective solution throughout the physical space of the annular blade row passage. The direct numerical implementation of the non-reflective conditions may lead to ill-conditioned matrices. An approach is proposed based on matrix factorization which enhances the convergence properties.