In this paper, a triple-band metamaterial absorber in the terahertz frequencies is proposed, and its refractive index sensing characteristics are analyzed, where the bulk Dirac semimetal (BDS) periodic array is on top of a photonic crystal slab backed with a metal ground plane. The simulation results show that the absorber achieves three perfect absorption peaks in the range of 3.4–5.2 THz, whose absorption rates are over 96%, and a maximum quality factor (Q) of 74.1. The designed absorber exhibits excellent polarization insensitivity and dynamic tunability; further, the tuning of the Fermi energy level of BDS enables the dynamic adjustment of absorption frequencies and absorption rates of these peaks. By analyzing the distributions of the electromagnetic field and different structural parameters, it is revealed that the absorber mainly dissipates the electromagnetic wave through coupled resonance and localized surface plasmon resonance (LSPR) effects to achieve perfect absorption. Further, the metamaterial absorber shows the capacity to detect analytes with varying refractive indices, and the absorber has a maximum sensitivity S of 405 GHz/RIU with high detection accuracy. This work provides novel design options for triple-band terahertz metamaterial absorbers and their potential applications in refractive index sensing.