Recent studies have shown that speciation through allopolyploidy, i.e., inter-specific or inter-generic hybridization followed by chromosome doubling, is accompanied by a variety of rapid cardinal genetic and epigenetic changes. This paper reviews our studies on the effect of allopolyploidization on several low-copy, non-coding sequences that exist in all the diploid species of the tribe Triticeae, including the progenitors of polyploid wheat, but in polyploid wheat they occur in only one genome, either in one homologous pair (chromosome-specific sequences) or in several pairs of the same genome (genome-specific sequences). Rapid elimination of these sequences from one genome is a general phenomenon in newly synthesized allopolyploids. Elimination was a nonrandom, reproducible event whose direction was determined by the genomic combination of the amphiploid. It was not affected by the genotype of the parental plants, by their cytoplasm, or by the ploidy level, and it did not result from intergenomic recombination. This elimination augmented the differentiation of homeologous chromosomes (partially homologous chromosomes of the different genomes) at the polyploid level, thus providing the physical basis for the diploid-like meiotic behavior characterizing polyploid wheat. This pattern of pairing prevents intergenomic recombination, and consequently, ensures full fertility, disomic inheritance, and permanent heterosis between alleles of different genomes (homeoalleles). Accordingly, rapid elimination of these sequences improves the fitness of newly formed allopolyploids, facilitating their rapid establishment in nature as new successful species.