<p>In this paper, we investigate the existence and bifurcation patterns of periodic traveling waves in a three-dimensional honeycomb lattice structure with nearest-neighbor interactions and nonlinear substrate potentials. The honeycomb lattice exhibits local rotational symmetry but lacks mirror symmetry. This unique feature necessitates the use of four distinct atoms for accurate modeling. By solving the coupled advanced-delay differential equations and employing the Lyapunov–Schmidt reduction method, we reduce the original problem to a finite-dimensional bifurcation equation that preserves the Hamiltonian structure. By combining invariant theory and singularity theory, we derive small-amplitude periodic solutions of the Hamiltonian system near equilibrium points in the resonant case. While most previous studies have only established the existence of bichromatic wave trains, we further demonstrate the existence of quadrichromatic wave trains and analyze their bifurcation patterns, which are influenced by two parameters.</p>