To realize the promise of 3D bioprinting, it is imperative to develop bioinks that possess the necessary biological and rheological characteristics for printing cell-laden tissue grafts. Alginate is widely used as a bioink because its rheological properties can be modified through pre-crosslinking or the addition of thickening agents to increase printing resolution. However, modification of alginate’s physiochemical characteristics using common crosslinking agents can affect its cytocompatibility. Therefore, we evaluated the printability, physicochemical properties, and osteogenic potential of four common alginate bioinks: alginate-CaCl2 (alg-CaCl2), alginate-CaSO4 (alg-CaSO4), alginate-gelatin (alg-gel) and alginate-nanocellulose (alg-ncel) for the 3D bioprinting of anatomically accurate osteogenic grafts. While all bioinks possessed similar viscosity, printing fidelity was lower in the pre-crosslinked bioinks. When used to print geometrically defined constructs, alg-CaSO4 and alg-ncel exhibited higher mechanical properties and lower mesh size than those printed with alg-CaCl2 or alg-gel. The physical properties of these constructs affected the biological performance of encapsulated bone marrow-derived mesenchymal stromal cells (MSCs). Cell-laden constructs printed using alg-CaSO4 and alg-ncel exhibited greater cell apoptosis and contained fewer living cells 7 days post-printing. In addition, effective cell-matrix interactions were only observed in alg-CaCl2 printed constructs. When cultured in osteogenic media, MSCs in alg-CaCl2 constructs exhibited increased osteogenic differentiation compared to the other three bioinks. This bioink was then used to 3D print anatomically accurate cell-laden scaphoid bones that were capable of partial mineralization after 14 days of in vitro culture. These results highlight the importance of bioink properties to modulate cell behavior and the biofabrication of clinically relevant bone tissues.