Three dimensional (3D) bioprinting is an emerging technology that enables complex spatial modeling of cell‐based tissue engineering products, whose therapeutic potential in regenerative medicine is enormous. However, its success largely depends on the definition of a bioprintable zone, which is specific for each combination of cell‐loaded hydrogels (or bioinks) and scaffolds, matching the mechanical and biological characteristics of the target tissue to be repaired. Therefore proper adjustment of the bioink formulation requires a compromise between: (i) the maintenance of cellular critical quality attributes (CQA) within a defined range of specifications to cell component, and (ii) the mechanical characteristics of the printed tissue to biofabricate. Herein, we investigated the advantages of using natural hydrogel‐based bioinks to preserve the most relevant CQA in bone tissue regeneration applications, particularly focusing on cell viability and osteogenic potential of multipotent mesenchymal stromal cells (MSCs) displaying tripotency in vitro, and a phenotypic profile of 99.9% CD105+/CD45,− 10.3% HLA‐DR,+ 100.0% CD90,+ and 99.2% CD73+/CD31− expression. Remarkably, hyaluronic acid, fibrin, and gelatin allowed for optimal recovery of viable cells, while preserving MSC's proliferation capacity and osteogenic potency in vitro. This was achieved by providing a 3D structure with a compression module below 8.8 ± 0.5 kPa, given that higher values resulted in cell loss by mechanical stress. Beyond the biocompatibility of naturally occurring polymers, our results highlight the enhanced protection on CQA exerted by bioinks of natural origin (preferably HA, gelatin, and fibrin) on MSC, bone marrow during the 3D bioprinting process, reducing shear stress and offering structural support for proliferation and osteogenic differentiation.