In this paper, we propose a new cubically convergent family of super-Halley method based on power means. Some well-known methods can be regarded as particular cases of the proposed family. New classes of higher (third and fourth) order multipoint iterative methods free from second order derivative are derived by semi-discrete modifications of above-mentioned methods. It is shown that super-Halley method is the only method which produces fourth order multipoint iterative methods. Furthermore, these multipoint methods with cubic convergence have also been extended for finding the multiple zeros of non-linear functions. Numerical examples are also presented to demonstrate the performance of proposed multipoint iterative methods.