  Let M be a compact locally conformal hyperkaehler manifold. We prove a
version of Kodaira-Nakano vanishing theorem for M. This is used to show that M
admits no holomorphic differential forms, and the cohomology of the structure
sheaf $H^i(O_M)$ vanishes for i>1. We also prove that the first Betti number of
M is 1. This leads to a structure theorem for locally conformally hyperkaehler
manifolds, describing them in terms of 3-Sasakian geometry. Similar results are
proven for compact Einstein-Weyl locally conformal Kaehler manifolds.
