We investigate the influence of crystallographic orientation and anisotropy on local phonon density of states, phonon transmissivity, and Kapitza conductance at interfaces between Lennard-Jones solids via classical molecular dynamics simulations. In agreement with prior works, we find that the Kapitza conductance at an interface between two face-centered cubic materials is independent of crystallographic orientation. On the other hand, at an interface between a face-centered cubic material and a tetragonal material, the Kapitza conductance is strongly dependent on the relative orientation of the tetragonal material, albeit this dependence is subject to the overlap in vibrational spectra of the cubic and tetragonal materials. Furthermore, we show that interactions between acoustic phonons in the cubic material and optical phonons in the tetragonal material can lead to the interface exhibiting greater “thermal anisotropy” as compared to that of the constituent materials. Finally, it is noted that the relative match or mismatch between the Debye temperatures of two materials comprising an interface does not serve an accurate gauge of the efficiency of interfacial thermal transport when those materials have different crystal structures.