Effective medium approximations are developed to estimate the conductivity of composite media that are random mixtures of inhomogeneities of various shapes. The approximations use dilute solution results for inclusions of certain shapes embedded in the effective medium matrix, and incorporate weight parameters to reflect the proportions of the inhomogeneities of various shapes from different component materials in a mixture. Since the approximations are based on the differential scheme, which is realizable by a certain hierarchical model, they never violate the mathematical requirements imposed upon a homogenized property including the bounds. As illustrations, the approximations are used to interpret empirical data on the electrical conductivity of water-saturated sedimentary rocks, and the thermal conductivity of some multiphase dispersions of metallic particles in silicone rubber matrix. (c) 2008 Published by Elsevier Ltd.