Multicellular aggregated tissues have grown critically important in benchtop biomedical research, both as stand-alone spheroids and when assembled into larger bioengineered constructs. However, typical systems for aggregate formation are limited in their capacity to reliably handle such cultures at various experimental stages in a broadly accessible, consistent, and scalable manner. In this work, we develop a broadly versatile all-in-one biofabrication strategy to form uniform, spherical, multicellular aggregates that can be maintained at precisely defined positions for analysis or transfer into a larger tissue. The 3D-printed MicroPocket Culture (MPoC) system consists of an array of simple geometry-based valves in a polyacrylamide hydrogel, and is able to produce hundreds of uniformly-sized aggregates in standard tissue culture well plates, using simple tools that are readily available in all standard biological wet-labs. The model breast cancer aggregates formed in these experiments are retained in defined positions on chip during all liquid handling steps required to stimulate, label, and image the experiment, enabling high-throughput studies on this culture model. Furthermore, MPoCs enable robust formation of aggregates in cell types that do not conventionally form such structures. Finally, we demonstrate that this single platform can also be used to generate complex 3D tissues from the precisely-positioned aggregate building blocks. To highlight the unique and broad versatility of this technique, we develop a simple 3D invasion assay and show that cancer cells preferentially migrate towards nearby model tumors; demonstrating the importance of spatial precision when engineering 3D tissues. Together, this platform presents a broadly accessible and uniquely capable system with which to develop advanced aggregate-based models for tissue engineering, fundamental research, and applied drug discovery.