The concentric rings two-dimensional (2D) k-space trajectory enables flexible trade-offs between image contrast, signal-to-noise ratio (SNR), spatial resolution, and scan time. However, to realize these benefits for in vivo imaging applications, a robust method is desired to deal with fat signal in the acquired data. Multipoint Dixon techniques have been shown to achieve uniform fat suppression with high SNR-efficiency for Cartesian imaging, but application of these methods for non-Cartesian imaging is complicated by the fact that fat off-resonance creates significant blurring artifacts in the reconstruction. In this work, two fat-water separation algorithms are developed for the concentric rings. A retracing design is used to sample rings near the center of k-space through multiple revolutions to characterize the fat-water phase evolution difference at multiple time points. This acquisition design is first used for multipoint Dixon reconstruction, and then extended to a spectroscopic approach to account for the trajectory's full evolution through 3D k-t space. As the trajectory is resolved in time, off-resonance effects cause shifts in frequency instead of spatial blurring in 2D k-space. The spectral information can be used to assess field variation and perform robust fat-water separation. In vivo experimental results demonstrate the effectiveness of both algorithms.