  We introduce a model for the condensate of dipolar atoms or molecules, in
which the dipole-dipole interaction (DDI) is periodically modulated in space,
due to a periodic change of the local orientation of the permanent dipoles,
imposed by the corresponding structure of an external field (the necessary
field can be created, in particular, by means of magnetic lattices, which are
available to the experiment). The system represents a realization of a nonlocal
nonlinear lattice, which has a potential to support various spatial modes. By
means of numerical methods and variational approximation (VA), we construct
bright one-dimensional solitons in this system, and study their stability. In
most cases, the VA provides good accuracy, and correctly predicts the stability
by means of the Vakhitov-Kolokolov (VK)\ criterion. It is found that the
periodic modulation may destroy some solitons, which exist in the usual setting
with unmodulated DDI, and can create stable solitons in other cases, not
verified in the absence of modulations. Unstable solitons typically transform
into persistent localized breathers. The solitons are often mobile, with
inelastic collisions between them leading to oscillating localized modes.
