Abstract Organ-on-a-chip systems that recapitulate tissue-level functions have been proposed to improve in vitro–in vivo correlation in drug development. Significant progress has been made to control the cellular microenvironment with mechanical stimulation and fluid flow. However, it has been challenging to introduce complex 3D tissue structures due to the physical constraints of microfluidic channels or membranes in organ-on-a-chip systems. Although this problem could be addressed with the integration of 3D bioprinting, it is not an easy task because the two technologies have fundamentally different fabrication processes. Inspired by 4D bioprinting, we develop a 4D subtractive manufacturing technique where a flexible sacrificial material can be patterned on a 2D surface, change shape when exposed to aqueous hydrogel, and subsequently degrade to produce perfusable networks in a natural hydrogel matrix that can be populated with cells. The technique is applied to fabricate organ-specific vascular networks, vascularized kidney proximal tubules, and terminal lung alveoli in a customized 384-well plate and then further scaled to a 24-well plate format to make a large vascular network, vascularized liver tissues, and for integration with ultrasound imaging. This biofabrication method eliminates the physical constraints in organ-on-a-chip systems to incorporate complex ready-to-perfuse tissue structures in an open-well design.