In recent years, microfluidic systems have been used to study fundamental aspects of angiogenesis through the patterning of single-layered, linear or geometric vascular channels. In vivo, however, capillaries exist in complex, three-dimensional (3D) networks, and angiogenic sprouting occurs with a degree of unpredictability in all x,y,z planes. The ability to generate capillary beds in vitro that can support thick, biological tissues remains a key challenge to the regeneration of vital organs. Here, we report the engineering of 3D capillary beds in an in vitro microfluidic platform that is comprised of a biocompatible collagen I gel supported by a mechanical framework of alginate beads. The engineered vessels have patent lumens, form robust ~1.5 mm capillary networks across the devices, and support the perfusion of 1 µm fluorescent beads through them. In addition, the alginate beads offer a modular method to encapsulate and co-culture cells that either promote angiogenesis or require perfusion for cell viability in engineered tissue constructs. This laboratory-constructed vascular supply may be clinically significant for the engineering of capillary beds and higher order biological tissues in a scalable and modular manner.