It is well-known that one cannot generally hear the shape of a drum: the metric of a compact surface is not uniquely determined by its Laplace-Beltrami spectrum. But one can still seek computational solutions to the inverse problem: given a sequence of eigenvalues, can we compute a surface whose Laplace-Beltrami spectrum approximates the sequence? I will discuss some numerical experiments related to this problem for the case of surfaces of sphere topology, whose discrete conformal parameterization leads to an especially simple formulation of the inverse problem.