There is significant interest within the tissue engineering and pharmaceutical industries to create 3D microphysiological systems of human organ function. The interest stems from a growing concern that animal models and simple 2D culture systems cannot replicate essential features of human physiology that are critical to predict drug response, or simply to develop new therapeutic strategies to repair or replace damaged organs. Central to human organ function is a microcirculation that not only enhances the rate of nutrient and waste transport by convection, but also provides essential additional physiological functions that can be specific to each organ. This review highlights progress in the creation of in vitro functional microvessel networks, and emphasizes organ-specific functional and structural characteristics that should be considered in the future mimicry of four organ systems that are of primary interest: lung, brain, liver, and muscle (skeletal and cardiac).