The emerging practice of data-driven storytelling is framing data using familiar narrative mechanisms such as slideshows, videos, and comics to make even highly complex phenomena understandable. However, current data stories are still not utilizing the full potential of the storytelling domain. One reason for this is that current data-driven storytelling practice does not leverage the full repertoire of media that can be used for storytelling, such as the spoken word, e-learning, and video games. In this paper, we propose a taxonomy focused specifically on media types for the purpose of widening the purview of data-driven storytelling simply by putting more tools into the hands of designers. Using our taxonomy as a generative tool, we also propose three novel storytelling mechanisms, including for live-streaming, gesture-driven oral presentations, and textual reports that dynamically incorporate visual representations.