Many animals use the visual motion generated by travelling straight, the translatory optic flow, to successfully navigate obstacles: near objects appear larger and to move more quickly than distant objects. Flies are expert at navigating cluttered environments, and while their visual processing of rotatory optic flow is understood in exquisite detail, how they process translatory optic flow remains a mystery. Here, we present novel cell types that have motion receptive fields matched to translation self-motion, the vertical translation (VT) cells. One of these, the VT1 cell, encodes forward sideslip self-motion, and fires action potentials in clusters - spike bursts. We show that the spike burst coding is size and speed-tuned, and is selectively modulated by motion parallax - the relative motion experienced during translation. These properties are spatially organized, so that the cell is most excited by clutter rather than isolated objects. When the fly is presented with a simulation of flying past an elevated object, the spike burst activity is modulated by the height of the object, and the single spike rate is unaffected. When the moving object alone is experienced, the cell is weakly driven. Meanwhile, the VT2-3 cells have motion receptive fields matched to the lift axis. In conjunction with previously described horizontal cells, the VT cells have properties well-suited to the visual navigation of clutter and to encode movements along near cardinal axes of thrust, lift and forward sideslip.