Abstract: Evolution equations with an underlying gradient flow structure have since long been of special interest in analysis and mathematical physics. In particular, transport equations that allow for a variational formulation with respect to the L2-Wasserstein metric have attracted a lot of attention recently. The gradient flow formulation gives rise to a natural semi-discretization in time of the evolution by means of the minimizing movement scheme, which constitutes a time-discrete minimization problem for the (sum of kinetic and potential) energy. On the other hand, nonlinear diffusion equations of fourth (and higher) order have become increasingly important in pure and applied mathematics. Many of them have been interpreted as gradient flows with respect to some metric structure. When it comes to solve equations of gradient flow type numerically, schemes that respect the equation's special structure are of particular interest. We present a fully discrete variant of the minimizing movement scheme for the numerical solution of the nonlinear fourth order Derrida-Lebowitz-Speer-Spohn equation in one space dimension, and discuss possible extensions to higher approximation order and to higher space dimensions.