The DMRadio suite of experiments uses lumped LC resonators to search for axions in lower frequency ranges than those accessible to cavity haloscopes. DMRadio-m$^3$ has sensitivity to DFSZ axions in the 30--200 MHz range corresponding to masses of $0.12\ \mu\text{eV}/c^2 < m_a < 0.83\ \mu\text{eV}/c^2$, and KSVZ axions in the 10--30 MHz range, corresponding to masses of $41\ \text{neV}/c^2 < m_a < 0.12\ \mu\text{eV}/c^2$ range. In this paper we present the design of DMRadio-m$^3$ as well as the method to extract the sensitivity of this experiment in a general form which applies to frequencies where the Compton wavelength of the axions is within a factor of order unity of the dimension of the detector. We also discuss the motivation and plan to use several coaxial pickup structures to cover the entire frequency range of the experiment.