We recently demonstrated a simple two-step approach for fabricating net-shaped metallic interpenetrating phase composites (IPCs) with tailorable thermal and mechanical properties. In the first step of this approach, selective laser melting is used to create a lattice preform. Next, this preform is liquid metal infiltrated with a second material that has a melting point lower than that of the lattice. Here we report on the processing-structure-thermal conductivity relations for these additively manufactured composites. Using experiments, periodic homogenization theory and finite element simulations, we show how the mesostructure, porosity, and properties of the constituents influence the effective thermal conductivity. The spatial distribution of porosity is found to strongly influence heat transport, suggesting unit cell geometries and processing strategies that will optimize the thermal conductivity.