Fourier transform techniques have been used to analyze the distributions of all ten independent DNA dinucleotide steps in two eukaryotic genomes and one prokaryotic genome, for periodicities of approximately 2 to 500 bp. The results reveal systematic deviations from random expectation for certain dinucleotide steps over this entire range of periodicities, together with striking peaks at certain spatial periodicities for particular dinucleotide steps. Several dinucleotides yield peaks at a periodicity of approximately 10.2 bp that are unique to the eukaryotic genomes. Certain members of this set of dinucleotide signals were previously identified as involved in nucleosome positioning, while others were previously unrecognized. In real-space, these dinucleotides are uncorrelated or even anticorrelated (relative to random expectation) at distances of 10 and 11 bp, despite having greater than random spectral power at the corresponding periodicity. Real-space correlations of these dinucleotides at distances of 10 and 11 bp are suppressed by another spectral component, a 3 bp periodicity attributed to codons, which has a local minimum probability at approximately 10.5 bp. When the two eukaryotic genomes are encoded for the signal "AA or TT", the peak at approximately 10.2 bp periodicity is strengthened, whereas for the prokaryotic genome such a peak remains absent. For the Caenorhabditis elegans genome, this peak becomes the dominant feature in the transform, surpassing a peak owing to the existence of codons in both height and integrated intensity. These results suggest that the requirements of chromosome structure place significant constraints on eukaryotic genome organization; they reveal additional signals that may be related to nucleosome positioning; and they reveal a wealth of additional new non-random aspects of genome sequence organization.