Achieving the ability to non-destructively, non-invasively examine subsurface features of living multicellular organisms at a microscopic level is currently a challenge for biologists. Optical coherence microscopy (OCM) is a new photonics-based technology that can be used to address this challenge. OCM takes advantage of refractive properties of biological molecules to generate three-dimensional images that can be viewed with a computer. We describe new data processing techniques and a different visualization algorithm that substantially improve OCM images. We have applied OCM imaging, in conjunction with these improvements, to a variety of structures of plants, including leaves, flowers, ovules and germinating seeds, and describe the visualization of cellular and subcellular structures within intact plants. We present evidence, based on detailed examination of our OCM images, comparisons to classical plant anatomy studies, and current knowledge of light scattering by cells and their components, that we can distinguish nuclei, organelles and vacuoles. Detailed examination of vascular tissue, which contains cells with elaborate wall structure, shows that cell walls produce no significant OCM signal. These improvements to the visualization process, together with the powerful non-invasive, non-destructive aspects of the technology, will broaden the application of OCM to questions in studies of plants as well as animals.