This is known, for projective (even Moishezon)
manifolds as shown by Dan Popovici in his
paper http://arxiv.org/abs/1003.3605
For general Kaehler manifold, this is conjectured.
Popovici has proved that a property of "strong Gauduchon"
is preserved in limits http://arxiv.org/abs/1009.5408
and (I think) there are no example of strong Gauduchon
manifold without Hodge decomposition.
The main obstruction to existence of Kahler metric (in addition to Lefschetz
SL(2)-action and Riemann-Hodge relations in cohomology)
is homotopy formality: the cohomology ring of a Kahler manifold is related to its de Rham algebra by a chain of homomorphisms of differential graded algebras inducing isomorphisms on cohomology. This is proven by Deligne-Griffiths-Morgan-Sullivan in 1970-ies.
This is a very strong topological condition; for instance, no nilmanifold (except torus) is homotopy formal. There are symplectic nilmanifolds satisfying hard Lefschetz and the rest of Riemann-Hodge conditions for cohomology.
Another obstruction is existence of a positive, exact current. As shown by Peternell, all non-Kahler Moishezon manifolds admit a positive, exact (n-1,n-1)-current, hence they are not Kahler. However, Moishezon manifolds are homotopy formal ([DGMS]), and often satisfy the Riemann-Hodge. This argument
is also used to prove that twistor spaces of compact Riemannian 4-manifolds are not Kahler, except CP^3 and flag space (Hitchin).
The sufficient condition in this direction is obtained by Harvey-Lawson: they proved that a manifold is Kahler if and only if it does not admit an exact (2n-2)-current with positive, non-zero (n-1, n-1)-part.
Finally, Izu Vaisman has shown that any compact locally conformally Kahler manifold (a manifold with Kahler metric taking values in a non-trivial 1-dimensional local system) is non-Kahler.
Also, a complex surface is Kahler if and only if its $b_1$ is even. This was known from Kodaira classification of surfaces, and the direct proof was obtained in late 1990-ies by Buchdahl and Lamari using the Harvey-Lawson criterion.
Best Answer
This question was debated in another forum a few years ago. The result was a note by Frédéric Campana in which he describes a counterexample as a corollary of another construction. In 1986 Gang Xiao (An example of hyperelliptic surfaces with positive index Northeast. Math. J. 2 (1986), no. 3, 255–257.) found two simply connected complex surfaces $S$ and $S'$ (that is, complex dimension 2), with different Hodge numbers, that are homeomorphic by Freedman's classification. The homeomorphism has to be orientation-reversing, but $S \times S$ and $S' \times S'$ are orientedly diffeomorphic and of course still have different Hodge numbers. Freedman's difficult classification is not essential to the argument, because in 8 real dimensions you can use standard surgery theory to establish the diffeomorphism.
Campana also explains that Borel and Hirzebruch found the first counterexample in 1959, in 5 complex dimensions.