Within the framework of the average approach and direct 3D PIC (particle-in-cell) simulations, we demonstrate that the gyrotrons operating in the regime of developed turbulence can sporadically emit "giant" spikes with intensities a factor of 100-150 greater than the average radiation power and a factor of 6-9 exceeding the power of the driving electron beams. Together with the statistical features such as a long-tail probability distribution, this allows the interpretation of generated spikes as microwave rogue waves. The mechanism of spikes formation is related to the simultaneous cyclotron interaction of a gyrating electron beam with forward and backward waves near the waveguide cutoff frequency as well as with the longitudinal deceleration of electrons.