In this paper a range-based relative localization solution is proposed and demonstrated in practice. The approach is based on wireless range measurements between robots, along with the communication of their velocities, accelerations, yaw rates, and height. It can be implemented on many robotic platforms without the need for dedicated sensors. With respect to previous work, we remove the dependency on a common heading reference between robots. The main advantage of this is that it makes the relative localization approach independent of magnetometer readings, which are notoriously unreliable in an indoor environment. A theoretical observability analysis shows that it may also have two disadvantages: the motion of the robots must meet more stringent conditions and the relative localization method becomes more susceptible to noise on the range measurements. However, simulation results have shown that in the presence of significant magnetic disturbances that are common to indoor environments, removing the heading dependency is beneficial. We conclude the paper by implementing the heading-independent method on real Micro Aerial Vehicles (MAVs) and performing leader-follower flight in an indoor environment. Despite the observability analysis showing leader-follower flight to be an especially difficult task, we still manage to successfully fly for over 3 minutes with two fully autonomous followers using only on-board sensing.