In this paper, we explore a few efficient linear solver and time discretization methods coupled with the local discontinuous Galerkin(LDG) spatial discretization to solve linear time-dependent fourth-order equations. We use implicit time discretization formulations to solve the difficulty of a unreasonably small time step for an explicit local time stepping method. The multigrid methods and domain decomposition method are introduced to solve the resulting discrete system of implicit methods. The Fourier analysis method are used to analyze the convergence behavior of the multigrid method. Comparisons are made among the multigrid solver and other solvers, to conclude that multigrid solver is efficient when coupled with the LDG spatial discretization for solving linear time-dependent fourth-order equations.