If L⊂ M is a Legendre submanifold in a Sasaki manifold, then the mean curvature flow does not preserve the Legendre condition. We define a kind of mean curvature flow for Legendre submanifolds which slightly differs from the standard one and then we prove that closed Legendre curves L in a Sasaki space form M converge to closed Legendre geodesics, if k²+σ+3≦ 0 and rot(L)=0, where σ denotes the sectional curvature of the contact plane ξ and k and rot(L) are the curvature respectively the rotation number of L. If rot(L)≠ 0, we obtain convergence of a subsequence to Legendre curves with constant curvature. In case σ+3≦ 0 and if the Legendre angle α of the initial curve satisfies osc(α) ≦ \pi, then we also prove convergence to a closed Legendre geodesic.