Current gold-standard strategies for bone regeneration do not achieve the optimal recovery of bone biomechanical properties. To bypass these limitations, tissue engineering techniques based on hybrid materials made up of osteoprogenitor cells—such as mesenchymal stem cells (MSCs)—and bioactive ceramic scaffolds—such as calcium phosphate-based (CaPs) bioceramics—seem promising. The biological properties of MSCs are influenced by the tissue source. This study aims to define the optimal MSC source and construct (i.e., the MSC–CaP combination) for clinical application in bone regeneration. A previous iTRAQ analysis generated the hypothesis that anatomical proximity to bone has a direct effect on MSC phenotype. MSCs were isolated from adipose tissue, bone marrow, and dental pulp, then cultured both on a plastic surface and on CaPs (hydroxyapatite and β-tricalcium phosphate), to compare their biological features. On plastic, MSCs isolated from dental pulp (DPSCs) presented the highest proliferation capacity and the greatest osteogenic potential. On both CaPs, DPSCs demonstrated the greatest capacity to colonise the bioceramics. Furthermore, the results demonstrated a trend that DPSCs had the most robust increase in ALP activity. Regarding CaPs, β-tricalcium phosphate obtained the best viability results, while hydroxyapatite had the highest ALP activity values. Therefore, we propose DPSCs as suitable MSCs for cell-based bone regeneration strategies.