Linear systems arising from implicit time discretizations and finite difference space discretizations of hyperbolic and parabolic equations in two–dimension are considered. A new circulant preconditioner is proposed for the solution of linear systems by the preconditioned conjugate gradient method. Our motivation is to exploit the fast inversion of circulant systems via the Fast Fourier Transform (FFT). For the second-order hyperbolic equations with initial and Dirichlet boundary conditions, we prove that the condition number of the preconditioned system is of O(mlogm), where m is the number of interior in each direction. The results can be extended to parabolic equations. Numerical results are also presented. Finally, we note that both the computation and the inversion of the circulant preconditioner and parallelized.