Assessing the statistical significance of structured RNA predicted from multiple sequence alignments relies on the existence of a good null model. We present here a random shuffling algorithm, Multiperm, that preserves not only the gap and local conservation structure in alignments of arbitrarily many sequences, but also the approximate dinucleotide frequencies. No shuffling algorithm that simultaneously preserves these three characteristics of a multiple (beyond pairwise) alignment has been available to date. As one benchmark, we show that it produces shuffled exonic sequences having folding free energy closer to native sequences than shuffled alignments that do not preserve dinucleotide frequencies.