Engineering organ-specific tissues for therapeutic applications is a grand challenge, requiring the fabrication and maintenance of densely cellular constructs composed of ~108 cells/ml. Organ building blocks (OBBs) composed of patient-specific-induced pluripotent stem cell-derived organoids offer a pathway to achieving tissues with the requisite cellular density, microarchitecture, and function. However, to date, scant attention has been devoted to their assembly into 3D tissue constructs. Here, we report a biomanufacturing method for assembling hundreds of thousands of these OBBs into living matrices with high cellular density into which perfusable vascular channels are introduced via embedded three-dimensional bioprinting. The OBB matrices exhibit the desired self-healing, viscoplastic behavior required for sacrificial writing into functional tissue (SWIFT). As an exemplar, we created a perfusable cardiac tissue that fuses and beats synchronously over a 7-day period. Our SWIFT biomanufacturing method enables the rapid assembly of perfusable patient- and organ-specific tissues at therapeutic scales.