We introduce a new Monte Carlo simulation technique for the equilibration of curvature-elastic interfacial systems such as surfactant films dissolved in bulk media. The method is based on a continuous representation of the interfaces and can accurately evaluate all relevant surface averages of the curvature Hamiltonian. We apply the method to sponge-like surfactant systems with the standard Hamiltonian, where the only parameters are the bending and saddle-splay moduli, kappa and kappaBAR. Random bicontinuous surface states are found to be stable for low bending rigidity and small negative saddle-splay modulus, justifying the use of Gaussian ensembles as approximations to real interfacial ensembles. Topological changes of the random sponge states as a function of the ratio kappaBAR/kappa are analyzed; the results over a wide range Of kappa suggest that only this universal ratio determines whether the final states resemble disordered minimal surfaces, disordered lamellar, or connected parabolic geometries.