  We study compact toric strict locally conformally K\"ahler manifolds. We show
that the Kodaira dimension of the underlying complex manifold is $-\infty$ and
that the only compact complex surfaces admitting toric strict locally
conformally K\"ahler metrics are the diagonal Hopf surfaces. We also show that
every toric Vaisman manifold has lcK rank 1 and is isomorphic to the mapping
torus of an automorphism of a toric compact Sasakian manifold.
