  We study the $\pm J$ three-dimensional Ising model with a longitudinal
anisotropic bond randomness on the simple cubic lattice. The random exchange
interaction is applied only in the $z$ direction, whereas in the other two
directions, $xy$ - planes, we consider ferromagnetic exchange. By implementing
an effective parallel tempering scheme, we outline the phase diagram of the
model and compare it to the corresponding isotropic one, as well as to a
previously studied anisotropic (transverse) case. We present a detailed
finite-size scaling analysis of the ferromagnetic - paramagnetic and spin glass
- paramagnetic transition lines, and we also discuss the ferromagnetic - spin
glass transition regime. We conclude that the present model shares the same
universality classes with the isotropic model, but at the symmetric point has a
considerably higher transition temperature from the spin-glass state to the
paramagnetic phase. Our data for the ferromagnetic - spin glass transition line
are supporting a forward behavior in contrast to the reentrant behavior of the
isotropic model.
