Three-dimensional (3D) bioprinting offers transformative potential for cardiac tissue engineering by enabling the fabrication of cell-laden constructs. However, key challenges remain, including maintaining cell viability within bioprinted constructs and understanding how embedded cells affect their physical and mechanical properties. This study addresses these challenges by incorporating human umbilical vein endothelial cells (HUVECs) into alginate–gelatin hydrogels and evaluating their impact on mechanical, physical, and rheological properties. Bioinks or hydrogels were prepared with or without HUVECs, and their rheological properties were assessed. Computational fluid dynamics (CFD) simulation was employed to determine the appropriate bioprinting pressure while minimizing cell damage. Constructs were designed and 3D-printed with an angular pattern to replicate the orientation of cardiac myofibrils and were characterized over a 21-day period for viscoelasticity, elastic modulus, swelling, mass loss, morphology, and cell viability. The incorporation of cells increased the storage and loss moduli of the bioink, demonstrating shear-thinning behavior as described by the Cross model. CFD simulation combined with preliminary cell viability assays identified 25 kPa as a suitable 3D-printing pressure, effectively preserving cell viability. Both cell-free and cell-laden constructs exhibited viscoelastic properties; however, cell-laden constructs displayed a lower elastic modulus under linear compression, reduced swelling, and greater mass retention. High cell viability was observed immediately post-bioprinting and was maintained for more than 1 week. These findings provide a framework for developing structurally robust, cell-laden constructs with enhanced functional fidelity, supporting their application in cardiac tissue engineering.