While biomaterials have become indispensable for a wide range of tissue repair strategies, second removal procedures oftentimes needed in the case of non-bio-based and non-bioresorbable scaffolds are associated with significant drawbacks not only for the patient, including the risk of infection, impaired healing, or tissue damage, but also for the healthcare system in terms of cost and resources. New biopolymers are increasingly being investigated in the field of tissue regeneration, but their widespread use is still hampered by limitations regarding mechanical, biological, and functional performance when compared to traditional materials. Therefore, a common strategy to tune and broaden the final properties of biopolymers is through the effect of different reinforcing agents. This research work focused on the fabrication and characterization of a bio-based and bioresorbable composite material obtained by compounding a poly(3-hydroxybutyrate-co-3-hydroxyhexanoate) (PHBH) matrix with acetylated cellulose nanocrystals (CNCs). The developed biocomposite was further processed to obtain three-dimensional scaffolds by additive manufacturing (AM). The 3D printability of the PHBH-CNC biocomposites was demonstrated by realizing different scaffold geometries, and the results of in vitro cell viability studies provided a clear indication of the cytocompatibility of the biocomposites. Moreover, the CNC content proved to be an important parameter in tuning the different functional properties of the scaffolds. It was demonstrated that the water affinity, surface roughness, and in vitro degradability rate of biocomposites increase with increasing CNC content. Therefore, this tailoring effect of CNC can expand the potential field of use of the PHBH biopolymer, making it an attractive candidate for a variety of tissue engineering applications.