Unpaired image-to-image translation has emerged as a crucial technique in medical imaging, enabling cross-modality synthesis, domain adaptation, and data augmentation without costly paired datasets. Yet, existing approaches often distort fine curvilinear structures, such as microvasculature, undermining both diagnostic reliability and quantitative analysis. This limitation is consequential in ophthalmic and vascular imaging, where subtle morphological changes carry significant clinical meaning. We propose Curvilinear Structure-preserving Translation (CST), a general framework that explicitly preserves fine curvilinear structures during unpaired translation by integrating structure consistency into the training. Specifically, CST augments baseline models with a curvilinear extraction module for topological supervision. It can be seamlessly incorporated into existing methods. We integrate it into CycleGAN and UNSB as two representative backbones. Comprehensive evaluation across three imaging modalities: optical coherence tomography angiography, color fundus and X-ray coronary angiography demonstrates that CST improves translation fidelity and achieves state-of-the-art performance. By reinforcing geometric integrity in learned mappings, CST establishes a principled pathway toward curvilinear structure-aware cross-domain translation in medical imaging.