Motivated by the theory of P -partitions, we introduce the basis of combinatorial power sums for the Hopf algebra of quasisymmetric functions, refining the power sum symmetric functions. These share many properties with the type 1 and type 2 quasisymmetric power sums introduced by Ballantine, Daugherty, Hicks, Mason and Niese, and moreover expand into the monomial basis of quasisymmetric functions with nonnegative integer coefficients. We prove formulas for products, coproducts and classical quasisymmetric involutions via the combinatorics of P -partitions, and give combinatorial interpretations for the coefficients when expanded into the monomial and fundamental bases. Contents 1. Introduction 1 2. Preliminaries 2 3. Weighted P -partitions 5 4. Quasisymmetric and noncommutative power sums 9 5. The basis of combinatorial power sums 10 References 16