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A new finite-difference scheme has been developed to solve efficiently the unsteady Euler equations for threedimensional inviscid supersonic flows with subsonic pockets. The technique utilizes planar Gauss-Seidel relaxation in the marching direction and approximate factorization in the crossflow plane. An "infinitely large" time step is used in parts of the flowfield where the component of velocity in the marching direction is supersonic-here the Gauss-Seidel sweeps are restricted to the forward direction only, and the procedure reduces to simple space-marching; a finite time step is used in parts of the flowfield where the marching component of velocity is subsonic-here, backward and forward Gauss-Seidel sweeps are employed to allow for upstream and downstream propagation of signals, and a time-asymptotic steady state is obtained. The discretization formulas are based on finite-volume implementations of high accuracy (up to third-order) total variation diminishing formulations. Numerical solutions are obtained for an analytically defined forebody, a realistic fighter configuration, and the Space Shuttle/The results are in very good agreement with available experimental data and numerical solutions of the full-potential equation.