Accurate segmentation of aortic dissection (AD) lumens in CT angiography (CTA) is essential for quantitative morphological assessment and clinical decision-making. However, reliable 3D delineation remains challenging due to limited long-range context modeling, which compromises inter-slice coherence, and insufficient structural discrimination under low-contrast conditions. To address these limitations, we propose BiM-GeoAttn-Net, a lightweight framework that integrates linear-time depth-wise state-space modeling with geometry-aware vessel refinement. Our approach is featured by Bidirectional Depth Mamba (BiM) to efficiently capture cross-slice dependencies and Geometry-Aware Vessel Attention (GeoAttn) module that employs orientation-sensitive anisotropic filtering to refine tubular structures and sharpen ambiguous boundaries. Extensive experiments on a multi-source AD CTA dataset demonstrate that BiM-GeoAttn-Net achieves a Dice score of 93.35% and an HD95 of 12.36 mm, outperforming representative CNN-, Transformer-, and SSM-based baselines in overlap metrics while maintaining competitive boundary accuracy. These results suggest that coupling linear-time depth modeling with geometry-aware refinement provides an effective, computationally efficient solution for robust 3D AD segmentation.