Eukaryotic cells are large enough to detect signals and then orient to them by differentiating the signal strength across the length and breadth of the cell. Amoebae, fibroblasts, neutrophils and growth cones all behave in this way. Little is known however about cell motion and searching behavior in the absence of a signal. Is individual cell motion best characterized as a random walk? Do individual cells have a search strategy when they are beyond the range of the signal they would otherwise move toward? Here we ask if single, isolated, Dictyostelium and Polysphondylium amoebae bias their motion in the absence of external cues. We placed single well-isolated Dictyostelium and Polysphondylium cells on a nutrient-free agar surface and followed them at 10 sec intervals for ~10 hr, then analyzed their motion with respect to velocity, turning angle, persistence length, and persistence time, comparing the results to the expectation for a variety of different types of random motion. We find that amoeboid behavior is well described by a special kind of random motion: Amoebae show a long persistence time (~10 min) beyond which they start to lose their direction; they move forward in a zig-zag manner; and they make turns every 1-2 min on average. They bias their motion by remembering the last turn and turning away from it. Interpreting the motion as consisting of runs and turns, the duration of a run and the amplitude of a turn are both found to be exponentially distributed. We show that this behavior greatly improves their chances of finding a target relative to performing a random walk. We believe that other eukaryotic cells may employ a strategy similar to Dictyostelium when seeking conditions or signal sources not yet within range of their detection system.