We consider fractional diffusion equations with discontinuous coefficients. By using an implicit finite difference scheme with the shifted Gr¨unwald discretization, the coefficient matrix of the resulting linear system can be composed as the sum of a scaled identity matrix and two diagonal-times-Toeplitz matrices. Due to the property of diffusion coefficients which are discontinuous, there are no circulant preconditioners that could work for such Toeplitz-like linear systems. The main idea of this paper is to propose and develop an approximate circulant inverse preconditioner for solving such discretized linear system with discontinuous coefficients. The construction is based on the piecewise linear interpolation to estimate eigenvalues of circulant matrices, but not to the diffusion coefficients. Thus characteristics of diffusion coefficient functions do not influence the construction of the proposed preconditioner, preserving the efficiency of the preconditioner. Theoretically, the spectra of the resulting preconditioned matrices are shown to be clustered around one. Numerical examples have demonstrated the effectiveness of the proposed preconditioner.