We report a detailed theoretical study of polymer nanorod composites using integral equation theory. The miscibility criteria for small amounts of nanorods in a dense polymer melt have been predicted for various polymer adsorption strengths, nanorod thicknesses, smoothnesses, and aspect ratios, for both athermally interacting and attractive rods. Close comparisons with the nanosphere limit have been made. We find that nanorods exhibit the same kinds of depletion, dispersion, bridging, and telebridging behavior previously predicted for spherical particles. However, the limits of these behaviors are a close (and sometimes nonmonotonic) function of aspect ratio, surface roughness, and nanorod thickness. The miscible region between depletion-driven phase separation and bridging driven phase separation quickly narrows with increasing aspect ratio and remains roughly constant as the nanorod radius of gyration exceeds that of the polymer. This miscible window narrows more quickly in the case of attractive rods. The polymer-mediated attractive forces between athermal rods scale with nanorod diameter, in both the bridging and depletion regimes. Telebridged structures are predicted when the range of polymerparticle attractions approaches the polymer thickness, i.e., the monomer diameter.