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Lattice materials provide unusual thermal and vibrational properties but not within the same structure. Thermal and vibrational multifunctionality is, however, crucial for thermomechanical applications such as automotive, aerospace, building, transportation, and energy infrastructure. In applications involving mobility, both high heat transfer and low mass are desired. Although there have been various efforts to design multifunctional lattice materials, the focus has largely remained on quasi‐static mechanical and thermal properties or mechanical and vibrational properties. Herein, designs of realizable lattice materials are reported, which are inherently thermally resistive, with vastly improved thermal conductance and omnidirectional phononic band gaps. By redesigning the truss structures to serve as interconnected heat pipes, a three‐order‐of‐magnitude improvement in the specific thermal conductance is found. Nodal masses at truss junctions are further used to obtain full vibrational band gaps. It is shown that it is possible to independently tune vibrational and thermal properties within the same structure. This work provides background for the design and fabrication of multifunctional lattice materials that simultaneously prevent structural vibrations and enhance heat conduction.