Music and rhythm-based training programs to improve reading are a novel approach to treatment of developmental dyslexia and have attracted the attention of trainers and researchers. Experimental studies demonstrating poor basic auditory processing abilities in individuals with dyslexia suggest they should be effective. On this basis, the efficacy of a novel rhythm-based intervention, Rhythmic Reading Training (RRT), was recently investigated and found to improve reading skills in Italian children with dyslexia, but its mode of action remains somewhat unclear. In this study, 19 children and preadolescents with dyslexia received 20 sessions of RRT over 10 weeks. Gains in a set of reading-related cognitive abilities—verbal working memory, auditory, and visual attention, and rhythm processing—were measured, along with reading outcomes. Analysis of the specific contribution of cognitive subprocesses to the primary effect of RRT highlighted that reading speed improvement during the intervention was related to rhythm and auditory discrimination abilities as well as verbal working memory. The relationships among specific reading parameters and the neuropsychological profile of participants are discussed.