We derive a precise motion law for fronts of solutions to scalar one-dimensional reaction-diffusion equations with multiple-wells, in the case the second derivative of the potential vanishes at its minimizers. We show that, renormalizing time in an algebraic way, the motion of fronts is governed by a simple system of ordinary differential equations of nearest neighbor interaction type. These interactions may be either attractive or repulsive. Our results are not constrained by the possible occurence of collisions nor splittings. They present substantial differences with the results obtained in the case the second derivative does not vanish at the wells, a case which has been extensively studied in the literature, and where fronts have been showed to move at exponentially small speed, with motion laws which are not renormalizable.