We investigate structured arrays and rings in elasticity to design elastic platonic circuits that utilise resonant phenomena. Creating ring resonators, and understanding their coupling to input and output arrays, allows for the development of platonic circuits including add–drop filters (ADFs) and coupled resonator elastic arrays (CREAs), and hence we envisage integrated platonic devices. Structured rings of point-masses placed atop a thin elastic plate lead to highly confined quasi-modes that leak energy; the leakage being quantified by the limiting quality factor Q. The conditions of resonance are deduced using highly accurate numerical simulations based on a Green’s function approach to solve the dispersion relation associated with the structured ring resonators. The sharp resonances that emerge are then used to filter and direct wave energy based on input frequency, and this is illustrated through analogy with devices used in optics, e.g. integrated photonic circuits. The potential applications of the elastic devices include elastic delay lines and passive energy harvesters.