Microwave tubes required to drive future 1-TeV linear colliders require 300-MW peak power at wavelengths near 3 cm. It is believed that gyrotrons have good potential for development into the high peak power, high gain, efficient amplifier required for this application. Presented here is the detailed design for a 3-cm gyroklystron amplifier at a peak power level of ~30 MW. This power level is approximately three orders of magnitude above the gyroklystron state of the art. Novel problems associated with a high-energy (γ~2) gyrotron beam are considered in detail. A new partially-self-consistent nonlinear analysis is described and used to design the gyroklystron circuit, which is based on the TE011 mode in cavities of circular cross section. Stable operation is anticipated.