We consider the system of linear equations L u = f, where L is a nonsymmetric block Toeplitz-like-plus-diagonal matrix, which arises from the Sinc-Galerkin discretization of differential equations. Our main contribution is to construct effective preconditioners for these structured coefficient matrices and to derive tight bounds for the eigenvalues of these preconditioned matrices. Moreover, we use numerical examples to show that the new preconditioner, when applied to the preconditioned GMRES method, is efficient for solving such kind of linear systems.