In this paper, a systematic demapping algorithm for three-dimensional (3-D) lattice signal constellations is presented. The algorithm consists of decision of an octant, computation of a distance from the origin, and determination of the coordinates of a symbol. Since the algorithm can be extended systematically, it is applicable to the larger lattice constellations. To verify the algorithm, 3-D signal transmission systems with field programmable gate array (FPGA) and $Matlab^{(R)}$ are implemented. And they are exploited to carry out computer simulation. As a result, both hardware and software based systems produce almost the same symbol error rates (SERs) in an additive white Gaussian noise (AWGN) environment. In addition, the hardware based system implemented with an FPGA generates waveforms of 3-D signals and recovers the original binary sequences perfectly. Those results confirm that the algorithm and the implemented 3-D transmission system operate correctly.