While the human body has many different examples of perfusable structures with complex geometries, biofabrication methods to replicate this complexity are still lacking. Specifically, the fabrication of self-supporting, branched networks with multiple channel diameters is particularly challenging. Here, we present the Gelation of Uniform Interfacial Diffusant in Embedded 3D Printing (GUIDE-3DP) approach for constructing perfusable networks of interconnected channels with precise control over branching geometries and vessel sizes. To achieve user-specified channel dimensions, this technique leverages the predictable diffusion of crosslinking reaction-initiators released from sacrificial inks printed within a hydrogel precursor. We demonstrate the versatility of GUIDE-3DP to be adapted for use with diverse physiochemical crosslinking mechanisms by designing seven printable material systems. Importantly, GUIDE-3DP allows for the independent tunability of both the inner and outer diameters of the printed channels and the ability to fabricate seamless junctions at branch points. This 3D bioprinting platform is uniquely suited for fabricating lumenized structures with complex shapes characteristic of multiple hollow vessels throughout the body. As an exemplary application, we demonstrate the fabrication of vasculature-like networks lined with endothelial cells. GUIDE-3DP represents an important advance toward the fabrication of self-supporting, physiologically relevant networks with intricate and perfusable geometries.