The application of cytocompatible hydrogels supporting extensive cellular activities to three-dimensional (3D) bioprinting is crucial for recreating complex physiological environments with high biomimicry. However, the poor printability and tunability of such natural hydrogels diminish the versatility and resolution of bioprinters. In this study, we propose a novel approach for the hybrid biofabrication of complex and heterogeneous 3D constructs using low-viscosity bioinks. Poly-(lactic acid) (PLA) filament is extruded by fused deposition modeling on a micromesh to create PLA-framed micromesh substrates onto which fibrinogen is printed by microextrusion bioprinting. The micromesh supports the printed hydrogel with a capillary pinning effect to enable high-resolution bioprinting. Accordingly, the micromesh-bioink layers are aligned and stacked to form volumetric constructs. This approach, called the 3D micromesh-bioink overlaid structure and interlocked culture (3D MOSAIC) platform, enables the fabrication of complicated and multimaterial 3D structures, including overhangs and voids. Endothelial cells cultured under vasculogenic conditions in the platform self-organize within the biologically functional hydrogel to form vascular networks, and cancer cell migration can be observed across the layers. The multidisciplinary 3D MOSAIC platform is an important step toward the biofabrication of complex constructs with high biological and structural significance and functionality.