See Claire Voisin's amazing results on the subject., or the published version:
On the homotopy types of compact kaehler and complex projective manifolds, Inventiones Math. 157 2 (2004), 329 - 343.
(ArXiv) Abstract: We show that in every dimension greater than or equal to 4, there exist compact Kaehler manifolds which do not have the homotopy type of projective complex manifolds. Thus they a fortiori are not deformation equivalent to a projective manifold, which solves negatively Kodaira's problem. We give both non simply connected (of dimension at least 4) and simply connected (of dimension at least 6) such examples.
