Lymphatic, nervous and tumor tissues exhibit complex physiology arising from three-dimensional interactions within genetically unique microenvironments. Here we develop a technology capable of volumetrically imaging transcriptomes, genotypes and morphologies in a single measurement, without relying on prior knowledge of spatial organization or genetic sequences. Our method extends DNA microscopy into three dimensions at scales involving 10⁷ molecules by forming a distributed intermolecular network of proximal unique DNA barcodes tagging complementary DNA molecules inside the specimen. After sequencing the DNA-encoded network, an image of molecular positions is inferred using geodesic spectral embeddings, a dimensionality reduction approach that we show to be especially suitable for this data-inverse problem. Applying whole-transcriptome volumetric DNA microscopy to intact zebrafish embryos, we demonstrate that three-dimensional image inference recapitulates zebrafish morphology and known gene expression patterns, capturing the spatial organization of gene sequences. Our extension of spatial genetic measurements to three dimensions, independent of prior templates, opens the door to detailed joint resolution of genomics and morphology in biological tissues.