Navigation in complex 3D scenarios requires appropriate environment representation for efficient scene understanding and trajectory generation. We propose a highly efficient and extensible global navigation framework based on a tomographic understanding of the environment to navigate ground robots in multi-layer structures. Our approach generates tomogram slices using the point cloud map to encode the geometric structure as ground and ceiling elevations. Then it evaluates the scene traversability considering the robot's motion capabilities. Both the tomogram construction and the scene evaluation are accelerated through parallel computation. Our approach further alleviates the trajectory generation complexity compared with planning in 3D spaces directly. It generates 3D trajectories by searching through multiple tomogram slices and separately adjusts the robot height to avoid overhangs. We evaluate our framework in various simulation scenarios and further test it in the real world on a quadrupedal robot. Our approach reduces the scene evaluation time by 3 orders of magnitude and improves the path planning speed by 3 times compared with existing approaches, demonstrating highly efficient global navigation in various complex 3D environments. The code is available at: https://github.com/byangw/PCT_planner.