Bartumeus et al. [Phys. Rev. Lett. 88, 097901 (2002)] compared the efficiency of a Lévy random walk predator strategy with a Brownian random walk strategy in a periodic one-dimensional domain with nonrevisitable moving targets. Their findings from numerical simulations conclude that "a Lévy search strategy is the best option in some, but not all, cases for a random search process." Using the same methodology, we show that the simplest random search strategy of all, ballistic motion in a random direction, outperforms a Lévy strategy in almost every case. We further show that, in the small set of cases where the ballistic strategy is not optimal, the periodic model does not capture the more realistic nonperiodic case. In the nonperiodic case, the ballistic strategy again outperforms the Lévy strategy.