In this paper, we exploit the Strang splitting technique for solving the multidimensional Allen-Cahn equations with anisotropic spatial fractional Riesz derivatives. Fully discrete numerical methods are proposed using exponential Strang's splitting schemes for the time integration with finite difference discretization in space. It is proved that the proposed methods can preserve the discrete maximum principle unconditionally. Furthermore, the fully discrete methods are theoretically confirmed to be convergent with second-order accuracy in both of time and space. In practical implementation, the proposed algorithms require to compute the matrix exponential associated with only one dimensional discretized matrices that possess Toeplitz structure. Meanwhile, a fast algorithm is further developed for evaluating the product of the Toeplitz matrix exponential with a vector. Numerical examples are presented to verify the theoretical analysis and demonstrate the efficiency of the proposed methods.