Research in animal models established that tinman, a key gene in Drosophila dorsal vessel development, is an orthologue of Nkx2-5, a key gene in vertebrate cardiac development. Similarities between the arthropod dorsal vessel and vertebrate hearts are interpreted in light of concepts such as homology or convergence. We discuss this controversy in the context of the evolution of animal circulatory pumps and propose the distinction between peristaltic and chambered pumps as a fundamental parameter for evolutionary comparisons between bilaterian pumps. Neither homology nor convergence is satisfactory to explain the origins of hearts and pumping organs. Instead, we propose that animal pumps derive from parallel improvements of an ancestral, peristaltic design represented by a layer of myocytes at the external walls of primitive vessels. This paradigm unifies disparate views, impacts our understanding of bilaterian evolution and may be helpful to interpret similarities between pumping organs of phylogenetically relevant species and emerging models.