A general radix-2 FFT algorithm was recently developed and implemented for Modern Single Instruction Multiple Data (SIMD) architectures. This algorithm (SIMD-FFT) was found to be faster than any scalar FFT implementation, and as well, than other FFT implementations that uses the SIMD architecture for complex 1D and 2D input data [1]. In this paper, the SIMD-FFT algorithm is extended to handle Multi-Dimensional input data; this new approach does not make use of matrix transposition. The results are compared against the FFTW for the 2D and 3D case. Overall, the SIMD-FFT was found to be faster for complex 2D input data (ranging from 82% up to 343%), and as well, for complex 3D input data (ranging from 59.5% up to 198%)