Accurate segmentation of 3D medical scans is crucial for clinical diagnostics and treatment planning, yet existing methods often fail to achieve both high accuracy and computational efficiency across diverse anatomies and imaging modalities. To address these challenges, we propose GCNV-Net, a novel 3D medical segmentation framework that integrates a Tri-directional Dynamic Nonvoid Voxel Transformer (3DNVT), a Geometrical Cross-Attention module (GCA), and Nonvoid Voxelization. The 3DNVT dynamically partitions relevant voxels along the three orthogonal anatomical planes, namely the transverse, sagittal, and coronal planes, enabling effective modeling of complex 3D spatial dependencies. The GCA mechanism explicitly incorporates geometric positional information during multi-scale feature fusion, significantly enhancing fine-grained anatomical segmentation accuracy. Meanwhile, Nonvoid Voxelization processes only informative regions, greatly reducing redundant computation without compromising segmentation quality, and achieves a 56.13% reduction in FLOPs and a 68.49% reduction in inference latency compared to conventional voxelization. We evaluate GCNV-Net on multiple widely used benchmarks: BraTS2021, ACDC, MSD Prostate, MSD Pancreas, and AMOS2022. Our method achieves state-of-the-art segmentation performance across all datasets, outperforming the best existing methods by 0.65% on Dice, 0.63% on IoU, 1% on NSD, and relatively 14.5% on HD95. All results demonstrate that GCNV-Net effectively balances accuracy and efficiency, and its robustness across diverse organs, disease conditions, and imaging modalities highlights strong potential for clinical deployment.