While the tunneling conductance between two spherical-like conducting particles depends on the relative interparticle distance, the wave function overlap between states of two rodlike particles, and so the tunneling conductance, depends also on the relative orientation of the rod axes. Modeling slender rodlike particles as cylindrical quantum wells of diameter D and length L >> D, we calculate the matrix element of the tunneling between two rods for arbitrary relative orientations of the rod axes. We show that tunneling between two parallel rods is about L/root D xi times larger than the tunneling matrix element for perpendicular rods, where xi is the tunneling decay length. By considering the full dependence of the tunneling conductance on the angle between rod axes, we calculate within an effective medium theory the conductivity of dispersions of rods with different degrees of alignment. We find that for isotropically oriented rods, the effect of orientation in the tunneling processes is marginal for all rod concentrations. On the contrary, for systems of strongly aligned rods, the enhanced tunneling between nearly parallel rods increases significantly the system conductivity in a relatively large concentration range. Next, we consider systems in which short-range attraction between rods is added, as in dispersions of rods with depletion interaction. We find that the strongly anisotropic attraction promotes enhanced tunneling between neighboring parallel rods, increasing the effective medium conductivity by several orders of magnitude compared to the case in which the angular dependence of tunneling is ignored, even for relatively weak attractions.