Let be the space of harmonic functions in the unit ball that are bounded by some increasing radial function that tends to infinity as r goes to one; these spaces are called growth spaces. We describe functions in growth spaces by the Cesàro means of their expansions in harmonic polynomials and apply this characterization to study coefficient multipliers between growth spaces. Further, we introduce spaces of harmonic functions of regular growth and show that integral operators considered recently in connection to boundary oscillation of harmonic functions in weighted spaces, can be realized as multipliers that map growth spaces to corresponding spaces of regular growth.