In this paper, we study the discretized linear systems arising from the space-fractional diffusion equations with piecewise continuous coefficients. Using the implicit finite difference scheme with the shifted Grünwald discretization, the resulting linear systems are Toeplitz-like which can be written as the sum of a scaled identity matrix and two diagonal-times-Toeplitz matrices. Standard circulant preconditioners and the existing approximate circulant-inverse preconditioner do not work for such Toeplitz-like linear systems since the discontinuous diffusion coefficients cannot be well approximated by interpolation polynomials. The main aim of this paper is to propose a new approximate circulant-inverse preconditioner to handle the fractional diffusion equations when the diffusion coefficients are piecewise continuous with finite jump discontinuities. Our idea is to approximate the eigenvalues of circulant matrices by the interpolation formula instead of approximating the diffusion coefficients as done by the existing algorithms. Therefore, the discontinuity of the diffusion coefficients does not influence the efficiency of the preconditioner. Theoretically, the spectra of the resulting preconditioned matrices are shown to be clustered around one, which can guarantee the fast convergence rate of the proposed preconditioner. Numerical examples are provided to demonstrate the effectiveness of our method.