A tunable tri-band terahertz refractive index sensor based on graphene is proposed and designed. The structure of the sensor comprises a gold (Au) substrate layer, a dielectric layer (Topas), and a graphene pattern layer. The numerical simulation results demonstrate that the sensor exhibits three narrowband absorption peaks at 1.68, 3.82, and 4.78 THz, and the corresponding absorption rate can reach 99.9%, 98.0%, and 97.9%, respectively. By adjusting the Fermi level of graphene, the resonant frequency of the sensor can be effectively tuned. Notably, due to the rotational symmetry of the structure, the sensor shows insensitivity to transverse magnetic and transverse electric polarizations of incident light. By varying the refractive index of surrounding analytes, the sensor’s sensitivity and Figure of Merit (FOM) are studied. The sensitivities of the three resonance peaks are 0.59, 1.19, and 1.38 THz/RIU, with corresponding FOM values of 2.57, 3.40, and 6.58, respectively. Furthermore, the feasibility and effectiveness of the designed sensor in practical applications are verified by testing five types of samples. These findings indicate that the proposed sensor has superior performance in sensitivity and selectivity, which makes it have a great potential for applications in the fields of material characterization, environmental monitoring, and biomedicine detection in the terahertz band.