Designing beampatterns with constant beamwidth over a wide range of frequencies is useful in many applications in speech, radar, sonar and communication. In this paper, we design constant-beamwidth beamformers for concentric ring arrays. The proposed beamformers utilize the circular geometry to provide improved beamwidth consistency compared to beamformers which are designed for linear sensor arrays of the same order. In the proposed configuration, all sensors on each ring share the same weight value. This constraint significantly simplifies the beamformers and reduces the hardware and computational resources required in a physical setup. Furthermore, a theoretical justification of the beamforming method is provided. We demonstrate the advantages of the proposed beamformers compared to the one-dimensional configuration in terms of directivity index, white noise gain and sidelobe attenuation.