This paper reports a polydimethylsiloxane microfluidic model system that can develop an array of nearly identical human microtissues with interconnected vascular networks. The microfluidic system design is based on an analogy with an electric circuit, applying resistive circuit concepts to design pressure dividers in serially-connected microtissue chambers. A long microchannel (550, 620 and 775 mm) creates a resistive circuit with a large hydraulic resistance. Two media reservoirs with a large cross-sectional area and of different heights are connected to the entrance and exit of the long microchannel to serve as a pressure source, and create a near constant pressure drop along the long microchannel. Microtissue chambers (0.12 μl) serve as a two-terminal resistive component with an input impedance > 50-fold larger than the long microchannel. Connecting each microtissue chamber to two different positions along the long microchannel creates a series of pressure dividers. Each microtissue chamber enables a controlled pressure drop of a segment of the microchannel without altering the hydrodynamic behaviour of the microchannel. The result is a controlled and predictable microphysiological environment within the microchamber. Interstitial flow, a mechanical cue for stimulating vasculogenesis, was verified by finite element simulation and experiments. The simplicity of this design enabled the development of multiple microtissue arrays (5, 12, and 30 microtissues) by co-culturing endothelial cells, stromal cells, and fibrin within the microchambers over two and three week periods. This methodology enables the culturing of a large array of microtissues with interconnected vascular networks for biological studies and applications such as drug development.