  In this article, we prove a weak type $(p,p)$ maximal inequality,
$1<p<\infty$, for weighted averages of a positive Dunford-Schwarz operator $T$
acting on a noncommutative $L_p$-space associated to a semifinite von Neumann
algebra $\mathcal{M}$, with weights in $W_q$, where
$\frac{1}{p}+\frac{1}{q}=1$. This result is then utilized to obtain modulated
individual ergodic theorems with $q$-Besicovitch and $q$-Hartman sequences as
weights. Multiparameter versions of these results are also investigated.
