As bioprinting advances into clinical relevance with patient-specific tissue and organ constructs, it must be capable of multi-material fabrication at high resolutions to accurately mimick the complex tissue structures found in the body. One of the most fundamental structures to regenerative medicine is microvasculature. Its continuous hierarchical branching vessel networks bridge surgically manipulatable arteries (~1-6 mm) to capillary beds (~10 µm). Microvascular perfusion must be established quickly for autologous, allogeneic, or tissue engineered grafts to survive implantation and heal in place. However, traditional syringe-based bioprinting techniques have struggled to produce perfusable constructs with hierarchical branching at the resolution of the arterioles (~100-10 µm) found in microvascular tissues. This study introduces the novel CEVIC bioprinting device (i.e., Continuously Extruded Variable Internal Channeling), a multi-material technology that breaks the current extrusion-based bioprinting paradigm of pushing cell-laden hydrogels through a nozzle as filaments, instead, in the version explored here, extruding thin, wide cell-laden hydrogel sheets. The CEVIC device adapts the chaotic printing approach to control the width and number of microchannels within the construct as it is extruded (i.e., on-the-fly). Utilizing novel flow valve designs, this strategy can produce continuous gradients varying geometry and materials across the construct and hierarchical branching channels with average widths ranging from 621.5 ± 42.92% µm to 11.67 ± 14.99% µm, respectively, encompassing the resolution range of microvascular vessels. These constructs can also include fugitive/sacrifical ink that vacates to leave demonstrably perfusable channels. In a proof-of-concept experiment, a co-culture of two microvascular cell types, endothelial cells and pericytes, sustained over 90% viability throughout 1 week in microchannels within CEVIC-produced gelatin methacryloyl-sodium alginate hydrogel constructs. These results justify further exploration of generating CEVIC-bioprinted microvasculature, such as pre-culturing and implantation studies.