3D occupancy prediction has recently emerged as a new paradigm for holistic 3D scene understanding and provides valuable information for downstream planning in autonomous driving. Most existing methods, however, are computationally expensive, requiring costly attention-based 2D-3D transformation and 3D feature processing. In this paper, we present a novel 3D occupancy prediction approach, H3O, which features highly efficient architecture designs that incur a significantly lower computational cost as compared to the current state-of-the-art methods. In addition, to compensate for the ambiguity in ground-truth 3D occupancy labels, we advocate leveraging auxiliary tasks to complement the direct 3D supervision. In particular, we integrate multi-camera depth estimation, semantic segmentation, and surface normal estimation via differentiable volume rendering, supervised by corresponding 2D labels that introduces rich and heterogeneous supervision signals. We conduct extensive experiments on the Occ3D-nuScenes and SemanticKITTI benchmarks that demonstrate the superiority of our proposed H3O.