Suppression of starlight by nulling interferometry has the potential to enable the direct detection of planets around nearby stars, and several interferometer configurations have been proposed which meet, to varying degrees, the numerous associated experimental goals, such as high sensitivity, high angular resolution, high stellar rejection, and extraneous signal rejection. However, the more capable approaches involve fairly complex systems, so that minimization of complexity is vital if a space-based nulling interferometry mission is to become a reality. The simplest case, the two-aperture nulling interferometer, possesses fairly limited capabilities, but most other configurations being considered for nulling interferometry missions involve four or more apertures. Another relatively simple case which has not received much attention to date is that of the three-aperture nulling interferometer. In this paper, the three-telescope case is discussed in detail, from the necessary triple-beam nulling beamcombiners, to the properties of the pupil plane configurations. It is concluded that three-telescope nullers share many of the desirable capabilities of four-telescope nullers, such as a phase modulation capability, and so in fact represent a very capable "simplest case" nulling interferometer configuration.