One popular and fruitful approach to understanding what influences the decision of where to look next has been to present targets in a series of trials either to the right or left of a central fixation point and examine sequential effects on saccadic latency. However, there is a problem with this paradigm: Every saccade to a target is necessarily followed by an equal and opposite movement back to the center, yet the potentially confounding influence of this refixation saccade is rarely considered. Here, we introduce a novel random-walk paradigm that eliminates this difficulty. Each successive target appears to the left or right of the previous one, allowing us to study long sequences of saccades uncontaminated by refixations. This exposes a new stimulus-history effect, which is remarkably prolonged and relates primarily to movement direction: A saccade reduces the latency for subsequent movements made in the same direction and retards those in the opposite direction. Although in conventional refixation paradigms this effect cancels out, it is of particular significance in the real world--where our fixation point shifts constantly with the object of interest--and reflects a prediction of the way that real objects typically move.