  In this paper, we study the boundedness theory for maximal Calder\'on-Zygmund
operators acting on noncommutative $L_p$-spaces. Our first result is a
criterion for the weak type $(1,1)$ estimate of noncommutative maximal
Calder\'on-Zygmund operators; as an application, we obtain the weak type
$(1,1)$ estimates of operator-valued maximal singular integrals of convolution
type under proper {regularity} conditions. These are the {\it first}
noncommutative maximal inequalities for families of linear operators that can
not be reduced to positive ones. For homogeneous singular integrals, the strong
type $(p,p)$ ($1<p<\infty$) maximal estimates are shown to be true even for
{rough} kernels.
  As a byproduct of the criterion, we obtain the noncommutative weak type
$(1,1)$ estimate for Calder\'on-Zygmund operators with integral regularity
condition that is slightly stronger than the H\"ormander condition; this
evidences somewhat an affirmative answer to an open question in the
noncommutative Calder\'on-Zygmund theory.
