We propose an efficient iterative method for generating random correlated binary sequences with a prescribed correlation function. The method is based on consecutive linear modulations of an initially uncorrelated sequence into a correlated one. Each step of modulation increases the correlations until the desired level has been reached. The robustness and efficiency of the proposed algorithm are tested by generating sequences with inverse power-law correlations. The substantial increase in the strength of correlation in the iterative method with respect to single-step filtering generation is shown for all studied correlation functions. Our results can be used for design of disordered superlattices, waveguides, and surfaces with selective transport properties.