In this paper, we theoretically and numerically demonstrate a tunable and polarization-dependent electromagnetically induced transparency analogy based on graphene terahertz metasurfaces. The unit cell of the metasurface consists of three-layer graphene strips embedded in a silicon grating. The dynamic adjustment of the transparent window can be achieved by changing the coupling distance between the graphene layers and the polarization direction of the incident lights. The operation mechanism behind the phenomenon can be attributed to the near-field interaction and electromagnetic coupling of modes in graphene strips. Furthermore, the full wave electromagnetic simulations obtained by the finite-difference time-domain method agree well with the theoretical fitting results based on the three-harmonic oscillator model. In addition, by changing the Fermi levels, it can not only realize the outstanding slow-light effects with a maximum group index of 3750 but also obtain the four-frequency asynchronous optical switch function in terahertz regions. Therefore, our proposed metamaterial device may have potential applications in image switching, optical switches, slow-light device, optical communication, and optical storage.