Autonomous aerial vehicles can provide efficient and effective solutions for radio frequency (RF) source tracking and localizing problems with applications ranging from wildlife conservation to search and rescue operations. Existing lightweight, low-cost, bearing measurements-based methods with a single antenna-receiver sensor system configurations necessitate in situ rotations, leading to substantial measurement acquisition times restricting searchable areas and number of measurements. We propose a GyroCopter for the task. Our approach plans the trajectory of a multi-rotor unmanned aerial vehicle (UAV) whilst utilizing UAV flight dynamics to execute a constant gyration motion to derive "pseudo-bearing" measurements to track RF sources. The gyration-based pseudo-bearing approach: i) significantly reduces the limitations associated with in situ rotation bearing; while ii) capitalizing on the simplicity, affordability, and lightweight nature of signal strength measurement acquisition hardware to estimate bearings. This method distinguishes itself from other pseudo-bearing approaches by eliminating the need for additional hardware to maintain simplicity, lightweightness and cost-effectiveness. To validate our approach, we derived the optimal rotation speed and conducted extensive simulations and field missions with our GyroCopter to track and localize multiple RF sources. The results confirm the effectiveness of our method, highlighting its potential as a practical and rapid solution for RF source localization tasks.