Artificially fabricated three-dimensional magnetic nanostructures have recently emerged as a new type of magnetic material with the potential of displaying physical properties absent in thin-film geometries. Interconnected nanowire arrays yielding three-dimensional versions of artificial spin-ices are of particular interest within this material category. Despite growing interest in the topic, several properties of these systems are still unexplored. Here, we study, through micromagnetic simulations, the high-frequency dynamic modes developing in buckyball-type magnetic nanoarchitectures. We obtain a characteristic excitation spectrum and analyze the corresponding mode profiles and their magnetic field dependence. The magnetic resonances are localized at different geometric constituents of the structure and depend on the local magnetic configuration. These features foreshow the potential of such systems for reprogrammable magnonic device applications with geometrically tunable frequencies.