  Axions are well motivated particles that could make up most or all of the
dark matter if they have masses below 100 $\mu$eV. Microwave cavity techniques
comprised of closed resonant structures immersed in solenoid magnets are
sensitive to dark matter axions with masses of a few $\mu$eV, but face
difficulties scaling to higher masses. We present the a novel detector
architecture consisting of an open, Fabry-P\'{e}rot resonator and a series of
current-carrying wire planes, and demonstrate this technique with a search for
dark matter axion-like particles called Orpheus. This search excludes dark
matter axion-like particles with masses between 68.2 and 76.5 $\mu$eV and
axion-photon couplings greater than $4\times10^{-7} \mathrm{GeV}^{-1}$. We
project that the fundamental sensitivity of this technique could be extended to
be sensitive to couplings below $1\times10^{-15} \mathrm{GeV}^{-1}$, consistent
with the DFSZ model of QCD axions.
