We report the generation of room temperature microwave oscillations from GaN/AlN resonant tunneling diodes, which exhibit record-high peak current densities. The tunneling heterostructure grown by molecular beam epitaxy on freestanding GaN substrates comprises a thin GaN quantum well embedded between two AlN tunneling barriers. The room temperature current-voltage characteristics exhibit a record-high maximum peak current density of ∼220 kA/cm2. When biased within the negative differential conductance region, microwave oscillations are measured with a fundamental frequency of ∼0.94 GHz, generating an output power of ∼3.0 μW. Both the fundamental frequency and the output power of the oscillator are limited by the external biasing circuit. Using a small-signal equivalent circuit model, the maximum intrinsic frequency of oscillation for these diodes is predicted to be ∼200 GHz. This work represents a significant step towards microwave power generation enabled by resonant tunneling transport, an ultra-fast process that goes beyond the limitations of current III-Nitride high electron mobility transistors.