In this paper we formulate the hydrodynamic moment and cumulant expansions of the density matrix for a quantum system in which the time evolution is described by an infinite hierarchy of coupled hydrodynamic moment equations. For the case where the system interacts with a complex environment, we develop a semiclassical closure that truncates the hierarchy. Our numerical approach for solving the truncated system of equations is intimately related to the de Broglie-Bohm interpretation of quantum mechanics, in that the time evolution of each cumulant is constructed over a discrete ensemble of hydrodynamic paths. We discuss the advantages and disadvantages of this scheme and apply the method to a model problem for a particle escaping from a metastable potential well while in contact with a thermal bath.