A grid-independent approximate Riemann solver has been developed for use in both two- and three-dimensional flows governed by the Euler or Navier- Stokes equations. Fluxes on grid faces are obtained via wave decomposition. By assuming that information propagates in the velocity- difference directions, rather than in the grid-normal directions as in a standard grid-aligned solver, this flux function more appropriately interprets and hence more accurately resolves shock and shear waves when these lie oblique to the grid.