A metamaterial-based terahertz sensor design is presented in this paper. The metamaterial unit structure comprises a graphene cross and three sets of graphene strips, which collectively achieve a triple plasmon-induced transparency (PIT) in both polarization directions. The theoretical results based on the Lorentz resonance theoretical model are in good agreement with finite difference time domain (FDTD) simulations. The sensor is polarization-independent and changes in the angle of incidence between 0° and 10° have a negligible impact on the sensing performance. Furthermore, the sensor exhibits a notable slow-light effect, with a maximum sensitivity of 1.65 THz/RIU and a FOM value of 4.74/RIU for the triple PIT, which is superior to that of comparable devices.