Recently, interest has been directed toward identifying and characterizing materials with 3D Dirac-like energy dispersions. Using a 3D version of the 2D Dirac-Weyl Hamiltonian, which has been used to describe the low energy physics of the 2D Dirac fermions found in graphene, we present our theoretical results for the electron self-energy of the 3D Dirac cone with the inclusion of an electron-phonon interaction (EPI). Employing a Holstein model for the EPI and allowing for varying chemical potential, bandwidth, and electron-phonon mass renormalization, we show how the self-energy modifies the electronic density of states and in turn the optical conductivity. The results for 3D are contrasted with the 2D case, as previously explored for graphene. Our results identify signatures of the EPI which can be used as a tool to understand experimental results probing the optical response and electronic properties of 3D analogues to the 2D Dirac fermions in graphene.