To cool a mechanical resonator to its ground-state is the key to realize the quantum manipulation of this mechanical resonator. We propose a cooling method based on electromagnetically-induced-transparency (EIT) and investigate the ground-state cooling of a mechanical resonator in a three-Laguerre-Gaussian-cavity optomechanical system, where two auxiliary cavities are coupled to the original one in the standard optomechanical system, respectively. When the optimal parameters are chosen, the optical fluctuation spectrum changes from Lorentzian shape to EIT-like one in a three-level atomic system. The asymmetry between cooling and heating rates makes it possible to realize the ground-state cooling of a mechanical resonator. The research results here provide a theoretical guidance for the cooling of mechanical resonators in three-Laguerre-Gaussian-cavity systems.