Recently we have developed a Monte Carlo algorithm for lattice spin systems that relies exclusively on the use of Fourier amplitudes without making any reference to the underlying real-space configurations. This article is devoted to give a comprehensive discussion of the motivation, basic ideas and limits of this algorithm. By analyzing its construction in detail, we explain why our algorithm is of considerable use for solving certain classes of problems that may be computationally too expensive otherwise. Also, we explain how finite size scaling is properly set up to work with the algorithm.