Microtubules are cytoskeletal filaments that play essential roles in many cellular processes and are key therapeutic targets in several diseases. Accurate segmentation of microtubule networks is critical for studying their organization and dynamics but remains challenging due to filament curvature, dense crossings, and image noise. We present MTFlow, a novel time-conditioned flow-matching model for microtubule segmentation. Unlike conventional U-Net variants that predict masks in a single pass, MTFlow learns vector fields that iteratively transport noisy masks toward the ground truth, enabling interpretable, trajectory-based refinement. Our architecture combines a U-Net backbone with temporal embeddings, allowing the model to capture the dynamics of uncertainty resolution along filament boundaries. We trained and evaluated MTFlow on synthetic and real microtubule datasets and assessed its generalization capability on public biomedical datasets of curvilinear structures such as retinal blood vessels and nerves. MTFlow achieves competitive segmentation accuracy comparable to state-of-the-art models, offering a powerful and time-efficient tool for filamentous structure analysis with more precise annotations than manual or semi-automatic approaches.