This paper develops two families of power method (PM) distributions based on polynomial transformations of the (1) Uniform, (2) Triangular, (3) Normal, (4) D-Logistic, and (5) Logistic distributions. One family is developed in the context of conventional method of moments and the other family is derived through the method of L-moments. As such, each of the five conventional moment-based PM classes has an analogous L-moment based class. A primary focus of the development is on PM polynomial transformations of order three. Specifically, systems of equations are derived for computing polynomial coefficients for user specified values of skew (L-skew) and kurtosis (L-kurtosis). Boundary regions for determining feasible combinations of skew (L-skew) and kurtosis (L-kurtosis) are also derived for determining if a set of solved coefficients yields a valid PM probability density function. Further, the conventional moment-based family of PM distributions is compared with its L-moment based analog in terms of estimation, power, outliers, and distribution fitting. The results of the comparison demonstrate that the L-moment based PM family is superior to the conventional moment-based family in each of the categories considered.