Predictive theory to geometrically engineer devices and materials in continuum systems to have desired topological-like effects is developed here by bridging the gap between quantum and continuum mechanical descriptions. A structured elastic plate, a bosoniclike system in the language of quantum mechanics, is shown to exhibit topological valley modes despite the system having no direct physical connection to quantum effects. We emphasize a predictive, first-principles, approach, the strength of which is demonstrated by the ability to design well-defined broadband edge states, resistant to backscatter, using geometric differences; the mechanism underlying energy transfer around gentle and sharp corners is described. Using perturbation methods and group theory, several distinct cases of symmetry-induced Dirac cones, which when gapped yield nontrivial band gaps, are identified and classified. The propagative behavior of the edge states around gentle or sharp bends depends strongly upon the symmetry class of the bulk media and we illustrate this via numerical simulations.