  A Sasakian manifold is a Riemannian manifold whose metric cone admits a
certain K\"ahler structure which behaves well under homotheties. We show that
the product of two compact Sasakian manifolds admits a family of complex
structures indexed by a complex nonreal parameter, none of whose members admits
any compatible locally conformally K\"ahler metrics if both Sasakian manifolds
are of dimension greater than $1$. We compare this family with another family
of complex structures which has been studied in the literature. We compute the
Dolbeault cohomology groups of these products of compact Sasakian manifolds.
