We present MeshTailor, the first mesh-native generative framework for synthesizing edge-aligned seams on 3D surfaces. Unlike prior optimization-based or extrinsic learning-based methods, MeshTailor operates directly on the mesh graph, eliminating projection artifacts and fragile snapping heuristics. We introduce ChainingSeams, a hierarchical serialization of the seam graph that prioritizes global structural cuts before local details in a coarse-to-fine manner, and a dual-stream encoder that fuses topological and geometric context. Leveraging this hierarchical representation and enriched vertex embeddings, our MeshTailor Transformer utilizes an autoregressive pointer layer to trace seams vertex-by-vertex within local neighborhoods, ensuring native coherent seams. Extensive evaluations show that MeshTailor produces more coherent, professional-quality seam layouts compared to recent optimization-based and learning-based baselines.