The goal of this paper is to efficiently solve Toeplitz systems with generating functions that are nonnegative, even and have zeroes in their range. The authors develop an idea previously examined by \textit{G. Fiorentino} and \textit{S. Serra} [ibid. 17, No. 5, 1068-1081 (1996; Zbl 0858.65039)] that consists in applying the multigrid method to those ill-conditioned systems. Both, the two-grid (TGM) and the multiple-grid (MGM) approaches are discussed. After proving convergence for TGM when using the damped-Jacobi method as smoothing operator, convergence for MGM is established as well. As to the computational cost, it is shown that it is \(O(n \log n)\) per iteration and about four times the cost involved in the circulant preconditioned conjugate gradient methods (CGM) or the band-Toeplitz preconditioned CGM. The examples examined show that convergence is achieved much faster than in the former CGM, yielding a lower computational cost. On the other hand, the overall computational cost appears to be higher when using MGM than when applying the band-Toeplitz CGM. However, it is pointed out that the convergence rate of CGM can be very much improved with other smoothing methods, such as red-black Jacobi, block-Jacobi and Gauss-Seidel. Moreover, there are functions for which MGM can be applied while band-Toeplitz CGM cannot. The results exposed in the paper show the interest of further developing the study of the MGM in order to numerically solve ill-conditioned Toeplitz systems.