  We compute residue currents of Bochner-Martinelli type associated with a
monomial ideal $I$, by methods involving certain toric varieties. In case the
variety of $I$ is the origin, we give a complete description of the annihilator
of the currents in terms of the associated Newton diagram. In particular, we
show that the annihilator is strictly included in $I$, unless $I$ is defined by
a complete intersection. We also provide partial results for general monomial
ideals.
