This paper is concerned with numerical methods for solving the multidimensional Allen-Cahn equations with spatial fractional Riesz derivatives. A fully discrete numerical scheme is proposed using a dimensional splitting exponential time differencing approximation for the time integration with finite difference discretization in space. Theoretically, we prove that the proposed numerical scheme can unconditionally preserve the discrete maximum principle. The error estimate in maximum-norm of the proposed scheme is also established in the fully discrete sense. In practical computation, the proposed algorithm can be carried out by computing linear systems and the matrix exponential associated with only one dimensional discretized matrices that possess Toeplitz structure. Meanwhile, fast methods for inverting the Toeplitz matrix and computing the Toeplitz exponential multiplying a vector are exploited to reduce the complexity. Numerical examples in two and three spatial dimensions are given to illustrate the effectiveness and efficiency of the proposed scheme.