  We prove a dimension-free $L^p(\mathbb{R}^d)$, $1<p<\infty$, estimate for the
vector of maximal Riesz transforms of odd order in terms of the corresponding
Riesz transforms. This implies a dimension-free $L^p(\mathbb{R}^d)$ estimate
for the vector of maximal Riesz transforms in terms of the input function. We
also give explicit estimates for the dependencies of the constants on $p$ when
the order is fixed. Analogous dimension-free estimates are also obtained for
single Riesz transforms of odd orders with an improved estimate of the
constants. These results are a dimension-free extension of the work of J.
Mateu, J. Orobitg, C. P\'erez, and J. Verdera. Our proof consists of
factorization and averaging procedures, followed by a non-obvious application
of the method of rotations.
