We determine, by first‐passage‐time simulations, the effective conductivity tensor&sgr;eof anisotropic suspensions of aligned spheroidal inclusions with aspect ratiob/a. This is a versatile model of composite media, containing the special limiting cases of aligned disks (b/a=0), spheres(b/a=1), and aligned needles (b/a=∞), and may be employed to model aligned, long‐ and short‐fiber composites, anisotropic sandstones, certain laminates, and cracked media. Data for&sgr;eare obtained for prolate cases (b/a=2, 5, and 10) and oblate cases (b/a=0.1, 0.2, and 0.5) over a wide range of inclusion volume fractions and selected phase conductivities (includingsuperconductinginclusions andperfectlyinsulating‘‘voids’’). The data always lie within second‐order rigorous bounds on&sgr;edue to Willis [J. Mech. Phys. Solids25, 185 (1977)] for this model. We compare our data for prolate and oblate spheroids to our previously obtained data for spheres [J. Appl. Phys.69, 2280 (1991)].