A new simulation technique is presented to efficiently yield the exact effective transport properties of disordered, statistically anisotropic, heterogeneous media in which the transport process is governed by a steady-state diffusion equation. The algorithm is one that improves over the first-passage-time technique developed by Torquato and Kim [1989, Appl. Phys. Lett. Vol.55, pp.1847-1849] that has been proved itself very efficient in terms of execution time. As an illustrative example, the trapping rate associated with diffusion-controlled-reaction process among static, aligned, spheroidal traps is computed. The algorithm is shown to have very fast execution time, typically an order of magnitude faster than the one by Torquato and Kim, still resulting in the exact values of trapping rate.