LiDAR odometry is a critical component of SLAM in autonomous driving and robotics. Learning-based methods have shown remarkable performance by regressing relative poses in an end-to-end manner. However, when applying these trained models, originally developed on the widely used KITTI dataset, to other scenes, performance often drops significantly. In other words, existing methods struggle to generalize well to new environments. To address this challenge, we propose RCP-LO, a simple yet effective LiDAR odometry framework. We introduce a novel representation for relative poses, reformulating them as relative coordinates, which can then be solved using geometrical verification. This approach avoids overly simplified pose representations and makes better use of scene geometry, thereby improving generalization. Moreover, to capture the inherent uncertainties in relative pose estimation from occluded LiDAR point clouds from dynamic environments, we adapt our framework to learn a denoising diffusion model, allowing for sampling plausible relative coordinates while improving robustness. We also introduce a differentiable geometric weighted singular value decomposition module, enabling efficient pose estimation through a single forward pass. Extensive experiments demonstrate that RCP-LO, trained exclusively on the KITTI dataset, achieves competitive performance compared to SOTA learning-based methods and generalizes effectively to the KITTI-360, Ford, and Oxford datasets. Our code will be made available upon acceptance.