In this work, we evaluated the influence of a novel hybrid 3D-printed porous composite scaffold based on poly(ε-caprolactone) (PCL) and β-tricalcium phosphate (β-TCP) microparticles in the process of adhesion, proliferation, and osteoblastic differentiation of multipotent adult human bone marrow mesenchymal stem cells (ah-BM-MSCs) cultured under basal and osteogenic conditions. The in vitro biological response of ah-BM-MSCs seeded on the scaffolds was evaluated in terms of cytotoxicity, adhesion, and proliferation (AlamarBlue Assay®) after 1, 3, 7, and 14 days of culture. The osteogenic differentiation was assessed by alkaline phosphatase (ALP) activity, mineralization (Alizarin Red Solution, ARS), expression of surface markers (CD73, CD90, and CD105), and reverse transcription–quantitative polymerase chain reaction (qRT-PCR) after 7 and 14 days of culture. The scaffolds tested were found to be bioactive and biocompatible, as demonstrated by their effects on cytotoxicity (viability) and extracellular matrix production. The mineralization and ALP assays revealed that osteogenic differentiation increased in the presence of PCL/β-TCP scaffolds. The latter was also confirmed by the gene expression levels of the proteins involved in the ossification process. Our results suggest that similar bio-inspired hybrid composite materials would be excellent candidates for osteoinductive and osteogenic medical-grade scaffolds to support cell proliferation and differentiation for tissue engineering, which warrants future in vivo research.