A muon cooling channel calls for very high accelerating gradient RF structures to restore the energy lost by muons in the absorbers. The RF structures have to be operated in a strong magnetic field and thus the use of superconducting RF cavities is excluded. To achieve a high shunt impedance while maintaining a large enough aperture to accommodate a large transverse emittance muon beam, the cavity design adopted is a pillbox-like geometry with thin Be foils to terminate the electromagnetic field at the cavity iris. The possibility of using grids of thin-walled metallic tubes for the termination is also being explored. Many of the RF-related issues for muon cooling channels are being studied both theoretically and experimentally using an 805 MHz cavity that has a pillbox-like geometry with thin Be windows to terminate the cavity aperture. The design and performance of this cavity are reported here. High-power RF tests of the 805 MHz cavity are in progress at Lab G in Fermilab. The cavity has exceeded its design gradient of 30 MV/m, reaching 34 MV/m without external magnetic field. No surface damage was observed at this gradient. The cavity is currently under conditioning at Lab G with an external magnetic field of 2.5 T. We also present here a 201 MHz cavity design for muoncooling channels. The proposed cavity design is also suitable for use in a proof-of-principle Muon Ionization Cooling Experiment (MICE).