We present 'Richardson-Lucy Network' (RLN), a fast and lightweight deep learning method for 3D fluorescence microscopy deconvolution. RLN combines the traditional Richardson-Lucy iteration with a fully convolutional network structure, improving network interpretability and robustness. Containing only ~16 thousand parameters, RLN enables 4- to 50-fold faster processing than purely data-driven networks with many more parameters. By visual and quantitative analysis, we show that RLN provides better deconvolution, better generalizability, and fewer artifacts than other networks, especially along the axial dimension. RLN outperforms Richardson-Lucy deconvolution on volumes contaminated with severe out of focus fluorescence or noise and provides 4- to 6-fold faster reconstructions of large, cleared tissue datasets than classic multi-view pipelines. We demonstrate RLN's performance on cells, tissues, and embryos imaged with widefield-, light sheet-, and structured illumination microscopy.