We perform N-body simulations of theories with infinite-volume extra dimensions, such as the Dvali- Gabadadze-Porrati model and its higher-dimensional generalizations, where 4D gravity is mediated by massive gravitons. The longitudinal mode of these gravitons mediates an extra scalar force, which we model as a density-dependent modification to the Poisson equation. This enhances gravitational clustering, particularly on scales that have undergone mild nonlinear processing. While the standard nonlinear fitting algorithm of Smith et al. overestimates this power enhancement on nonlinear scales, we present a modified fitting formula that offers a remarkably good fit to our power spectra. Because of the uncertainty in galaxy bias, our results are consistent with precision power spectrum determinations from galaxy redshift surveys, even for graviton Compton wavelengths as small as 300 Mpc. Our model is sufficiently general that we expect it to capture the phenomenology of a wide class of related higher-dimensional gravity scenarios.