Most of the studies in three-dimensional (3D) bioprinting have been traditionally based on printing a single bioink. Addressing the complexity of organ and tissue engineering, however, will require combining multiple building and sacrificial biomaterials and several cells types in a single biofabrication session. This is a significant challenge, and, to tackle that, we must focus on the complex relationships between the printing parameters and the print resolution. In this paper, we study the influence of the main parameters driven multi-material 3D bioprinting and we present a method to calibrate these systems and control the print resolution accurately. Firstly, poloxamer hydrogels were extruded using a desktop 3D printer modified to incorporate four microextrusion-based bioprinting (MEBB) printheads. The printed hydrogels provided us the particular range of printing parameters (mainly printing pressure, deposition speed, and nozzle z-offset) to assure the correct calibration of the multi-material 3D bioprinter. Using the printheads, we demonstrated the excellent performance of the calibrated system extruding different fluorescent bioinks. Representative multi-material structures were printed in both poloxamer and cell-laden gelatin-alginate bioinks in a single session corroborating the capabilities of our system and the calibration method. Cell viability was not significantly affected by any of the changes proposed. We conclude that our proposal has enormous potential to help with advancing in the creation of complex 3D constructs and vascular networks for tissue engineering.