Salt fingers occur throughout a large fraction of the World Ocean and can have substantial effects on large-scale mixing processes, such as the Meridional Overturning Circulation (see, e.g., Zhang et al., 1998). However, most numerical and laboratory studies of this phenomenon occur in quiescent environments. We simulate salt fingers in the presence of constant and oscillating shear in order to quantify the mixing of heat and salt by these systems under the impacts of large-scale internal waves. The code used in these simulations (the “Rocking Ocean Modeling Environment” or ROME) is a new pseudo-spectral hydrodynamic model which incorporates a steady or oscillatory background shear flow with a spatially uniform background velocity gradient. This configuration presents a challenge for modeling via Fourier-based algorithms because the typical evolution of such a flow is incompatible with the periodic boundary conditions at the vertical extremities of the computational domain. This complication is addressed by reformulating the governing equations in a new, temporally varying “tilting” coordinate system associated with the background flow as has been done in the past in the field of homogenous turbulence. Generally, it is shown that the application of shear can reduce fluxes by a factor of 2 or 3 for typical amplitudes of near-inertial waves and that the impact of shear decreases as the frequency of the applied shear increases. Though the focus of this study is on the effects of shear on double-diffusive systems, ROME is well-suited to a wide range of problems involving sheared stratified systems.