Narrative visualization aims to communicate scientific results to a general audience and garners significant attention in various applications. Merging exploratory and explanatory visualization could effectively support a non-expert understanding of scientific processes. Medical research results, e.g., mechanisms of the healthy human body, explanations of pathological processes, or avoidable risk factors for diseases, are also interesting to a general audience that includes patients and their relatives. This paper discusses how narrative techniques can be applied to medical visualization to tell data-driven stories about diseases. We address the general public comprising people interested in medicine without specific medical background knowledge. We derived a general template for the narrative medical visualization of diseases. Applying this template to three diseases selected to span bone, vascular, and organ systems, we discuss how narrative techniques can support visual communication and facilitate understanding of medical data. Other scientists can adapt our proposed template to inform an audience on other diseases. With our work, we show the potential of narrative medical visualization and conclude with a comprehensive research agenda.