We study perturbative unitarity in the scalar sector of the Myers-Pospelov model. The model introduces a preferred four-vector <math><mi>n</mi></math> which breaks Lorentz symmetry and couples to a five-dimension operator. When the preferred four-vector is chosen in the pure timelike or lightlike direction, the model becomes a higher time derivative theory, leading to a cubic dispersion relation. Two of the poles are shown to be perturbatively connected to the standard ones, while a third pole, which we call the Lee-Wick-like pole, is associated to a negative metric in Hilbert space, threatening the preservation of unitarity. The pure spacelike case is a normal theory in the sense that it has only two solutions, both being small perturbations over the standard ones. We analyze perturbative unitarity for purely spacelike and timelike cases using the optical theorem and considering a quartic self-interaction term. By computing discontinuities in the loop diagram, we arrive at a pinching condition which determines the propagation of particles and Lee-Wick-like particles through the cut. We find that the contribution for Lee-Wick-like particles vanishes for any external momenta, leaving only the contribution of particles, thus preserving one-loop unitarity in both cases.