A dynamically wavelength tunable multispectral plasmon induced transparency (PIT) device based on graphene metamaterials, which is composed of periodically patterned graphene double layers separated by a dielectric layer, is proposed theoretically and numerically in the terahertz frequency range. Considering the near-field coupling of different graphene layers and the bright-dark mode coupling in the same graphene layer, the coupled Lorentz oscillator model is adapted to explain the physical mechanism of multispectral EIT-like responses. The simulated transmission based on the finite-difference time-domain (FDTD) solutions indicates that the shifting and depth of the EIT resonances in multiple PIT windows are controlled by different geometrical parameters and Fermi energies distributions. A design scheme with graphene integration is employed, which allows independent tuning of resonance frequencies by electrostatically changing the Fermi energies of graphene double layer. Active control of the multispectral EIT-like responses enables the proposed device to be widely applied in optical information processing as tunable sensors, switches, and filters.