Organoids, the 3D culture systems derived from stem cells, are promising models for human organs. However, organoid study requires large-volume imaging with single cell resolution, which is beyond the spatial bandwidth limit of conventional optical microscopy. Herein, we propose a lensless holographic microscope empowered with a time and memory-saving algorithm. It solves the trade-off between the imaging field of view, resolution, and processing speed, and provides a practical tool for the study of organoids. We first build a compact microscopy system using a multi-angle LED illumination scheme and an on-chip structure. Then we develop a fast angular spectrum formula for fast reconstruction of oblique-illuminated coaxial holography under the under-sampling condition. Additionally, we derive a multi-angle illuminated filtered backpropagation algorithm to achieve high-precision and slice-wise recovery of 3D structures of objects. The reconstruction process demands only 1/50 of the memory required by a traditional optical diffraction tomography algorithm. Experimental results indicate that the proposed method can achieve 6.28 mm × 4.71 mm × 0.37 mm volume imaging within 104 s. Through the standardized polystyrene beads test, we demonstrate that the proposed microscope has micrometer-scale resolution in both lateral and axial directions. In addition, the 3D imaging results of salivary gland organoids show great application prospects of the proposed method in the field of living biological sampling imaging.