  Let $P$ be a set of $n$ points in $\mathbb{R}^3$ in general position, and let
$RCH(P)$ be the rectilinear convex hull of $P$. In this paper we obtain an
optimal $O(n\log n)$-time and $O(n)$-space algorithm to compute $RCH(P)$. We
also obtain an efficient $O(n\log^2 n)$-time and $O(n\log n)$-space algorithm
to compute and maintain the set of vertices of the rectilinear convex hull of
$P$ as we rotate $\mathbb R^3$ around the $z$-axis. Finally we study some
properties of the rectilinear convex hulls of point sets in $\mathbb{R}^3$.
