The common use of hydrogel materials in 3D bioprinting techniques is dictated by the unique properties of hydrogel bioinks, among which some of the most important in terms of sustaining vital cell functions in vitro in 3D cultures are the ability to retain large amounts of liquid and the ability to modify rigidity and mechanical properties to reproduce the structure of the natural extracellular matrix. Due to their high biocompatibility, non-immunogenicity, and the possibility of optimizing rheological properties and bioactivity at the same time, one of the most commonly used hydrogel bioink compositions are polymer solutions based on sodium alginate and gelatin. In 3D bioprinting techniques, it is necessary for hydrogel printouts to feature an appropriate geometry to ensure proper metabolic activity of the cells contained inside the printouts. The desired solution is to obtain a thin-walled printout geometry, ensuring uniform nutrient availability and gas exchange during cultivation. Within this study’s framework, tubular bioprinted structures were developed based on sodium alginate and gelatin, containing cells of the immortalized fibroblast line NIH/3T3 in their structure. Directly after the 3D printing process, such structures are characterized by extremely low mechanical strength. The purpose of this study was to perform a comparative analysis of the viability and spreading ability of the biological material contained in the printouts during their incubation for a period of 8 weeks while monitoring the effect of cellular growth on changes in the mechanical properties of the tubular structures. The observations demonstrated that the cells contained in the 3D printouts reach the ability to grow and spread in the polymer matrix after 4 weeks of cultivation, leading to obtaining a homogeneous, interconnected cell network inside the hydrogel after 6 weeks of incubation. Analysis of the mechanical properties of the printouts indicates that with the increasing time of cultivation of the structures, the degree of their overgrowth by the biological material contained inside, and the progressive degradation of the polymer matrix process, the tensile strength of tubular 3D printouts varies.