  This paper is devoted to the detailed analysis of three-dimensional motions
in $\mathbb{R}^3$ with orthogonal directions switching at Poisson times and
moving with constant speed $c>0$. The study of the random position at an
arbitrary time $t>0$ on the surface of the support, forming an octahedron
$S_{ct}$, is completely carried out on the edges $E_{ct}$ and faces $F_{ct}$.
In particular, the motion on the faces $F_{ct}$ is analysed by means of a
transformation which reduces it to a three-directions planar random motion.
This permits us to obtain an integral representation on $F_{ct}$ in terms of
integral of products of first order Bessel functions. The investigation of the
distribution of the position $p=p(t,x,y,z)$ inside $S_{ct}$ implied the
derivation of a sixth-order partial differential equation governing $p$
(expressed in terms of the products of three D'Alembert operators). A number of
results, also in explicit form, concern the time spent on each direction and
the position reached by each coordinates as the motion devolpes. The analysis
is carried out when the incoming direction is orthogonal to the ongoing one and
also when all directions can be uniformely choosen at each Poisson event. If
the switches are governed by homogeneus Poisson process many explicit results
are obtained.
