As engineered tissues progress toward therapeutically relevant length scales and cell densities, it is critical to deliver oxygen and nutrients throughout the tissue volume via perfusion through vascular networks. Furthermore, seeding of endothelial cells within these networks can recapitulate the barrier function and vascular physiology of native blood vessels. In this protocol, we describe how to fabricate and assemble customizable open-source tissue perfusion chambers and catheterize tissue constructs inside them. Human endothelial cells are seeded along the lumenal surfaces of the tissue constructs, which are subsequently connected to fluid pumping equipment. The protocol is agnostic with respect to biofabrication methodology as well as cell and material composition, and thus can enable a wide variety of experimental designs. It takes ~14 h over the course of 3 d to prepare perfusion chambers and begin a perfusion experiment. We envision that this protocol will facilitate the adoption and standardization of perfusion tissue culture methods across the fields of biomaterials and tissue engineering.