We investigate sequences of complex numbers $a = a_k$ for which the modulated averages $1nn_k=1a_kT^k f$ converge in norm for every weakly almost periodic linear operator $T$ in a Banach space. For Dunford-Schwartz operators on probability spaces, we study also the a.e. convergence in $L_p$. The limit is identified in some special cases, in particular when $T$ is a contraction in a Hilbert space, or when $a = S^k$ for some positive Dunford-Schwartz operator $S$ on a Lebesgue space and $phi 2$. We also obtain necessary and sufficient conditions on $a$ for the norm convergence of the modulated averages for every mean ergodic power bounded $T$, and identify the limit.