An analytic integral equation treatment is developed for the entropic depletion attractions induced by flexible polymers in dilute solutions of rigid rod-like macromolecules. The polymer-induced changes in the pair correlation functions between rod-like molecules and the rod-rod second virial coefficient are compared with results derived from a scaling approach. Results from both approaches predict that, for flexible coil-like depletants with radii of gyration sufficiently large compared to the rod lengths, the rod-rod second virial coefficient shows a nonmonotonic dependence on the coil concentration with a minimum located near the semidilute threshold. Our results are analogous to prior findings for athermal mixtures of flexible coils and hard spheres, and the predicted nonmonotonicities in the second virial coefficients between dispersed particles have the same physical origins. (C) 2003 American Institute of Physics.