This article is concerned with the minimal wave speed of traveling wave solutions for an integrodifference equation of higher order. Besides the operator may be nonmonotone, the kernel functions may be not Lebesgue measurable and integrable such that the equation has lower regularity. By constructing a proper set of potential wave profiles, we obtain the existence of smooth traveling wave solutions when the wave speed is larger than a threshold. Here, the profile set is obtained by giving a pair of upper and lower solutions. When the wave speed is the threshold, the existence of nontrivial traveling wave solutions is proved by passing to a limit function. Moreover, we obtain the nonexistence of nontrivial traveling wave solutions when the wave speed is smaller than the threshold.