Riemannian wavefield extrapolation (RWE) is used to model one-way wave propagation on generalized coordinate meshes. Previous RWE implementations assume that coordinate systems are defined by either orthogonal or semiorthogonal geometry. This restriction leads to situations where coordinate meshes suffer from problematic bunching and singularities. Nonorthogonal RWE is a procedure that avoids many of these problems by posing wavefield extrapolation on smooth, but generally nonorthogonal and singularity-free, coordinate meshes. The resulting extrapolation operators include additional terms that describe nonorthogonal propagation. These extra degrees of complexity, however, are offset by smoother coefficients that are more accurately implemented in one-way extrapolation operators. Remaining coordinate mesh singularities are then eliminated using a differential mesh smoothing procedure. Analytic extrapolation examples and the numerical calculation of 2D and 3D Green's functions for cylindrical and near-spherical geometry validate the nonorthogonal RWE propagation theory. Results from 2D benchmark testing suggest that the computational overhead associated with the RWE approach is roughly 35% greater than Cartesian-based extrapolation.