An efficient and powerful full-wave electromagnetic technique is presented to characterise and design periodic metamaterial structures. First, the spectral finite-difference time-domain (FDTD) method with periodic boundary conditions and uniaxial perfect matched layer is employed to predict the performance of a mushroom-like artificial magnetic conductor (AMC) surface and further extended to characterise a negative-refractive-index material consisting of lumped and distributed transmission-line elements. Then, a new computational technique is developed to design and optimise periodic metamaterial structures by integrating the spectral FDTD method with a genetic algorithm (GA), namely the micro-genetic algorithm. This computational technique is successfully applied to design and optimise single-band and dual-band AMC structures consisting of a frequency-selective surface and a ground plane. It is demonstrated that the GA/FDTD technique is a very effective approach for the design and optimisation of periodic metamaterial structures consisting of dielectrics and conductors of arbitrary configurations.