Lattices based on triply periodic minimal surfaces (TPMSs), which have been receiving increasing interest due to advances in additive manufacturing, are known now to be outperforming other cellular materials in several properties, enabling wider multifunctional applications. In this work, we focus on fabricating graphene TPMS lattices, viz., Gyroid, Schoen's I-WP (IWP), and Diamond, and investigate their multifunctional properties experimentally and computationally. The three-dimensional (3D) graphene structures were synthesized using a self-assembly hydrothermal-assisted dip-coating technique and the resulting lattices were tested for their mechanical, thermal, and electrical properties and compared to finite element simulation results. The graphene TPMS lattices resulting from the fabrication take the topology of a new class of TPMS architectures that are referred to here as the TPMS tube-networks. IWP demonstrated the highest elastic modulus as well as electrical and thermal conductivities. This study shows that 3D porous tube graphene can be utilized in designing new lightweight structural materials of low density with controllable thermal and electrical properties and mechanical strength with a potential to be employed in multifunctional engineering applications.