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.
Every compact complex manifold satisfying the $\partial\overline{\partial}$-Lemma has such a property. Particular examples are given by - as you already said - Hironaka example (and, more in general, Moishezon manifolds and manifolds in class C of Fujiki), or some deformations of twistor spaces (see LeBrun, Poon, Twistors, Kähler manifolds, and bimeromorphic geometry. II, J. Amer. Math. Soc. 5 (1992), no. 2, 317–325).
On the other side: in Ceballos, Otal, Ugarte, Villacampa, arXiv:1111.5873, Proposition 4.3, you find a concrete example of a compact complex manifold with the symmetry of Hodge diamond you require. This example does not satisfy $\partial\overline{\partial}$-Lemma.
Best Answer
Let $X$ and $Y$ be compact complex manifolds. Note that $K_{X\times Y} \cong \pi_1^*K_X\otimes \pi_2^*K_Y$. If $Y$ has trivial canonical bundle, then $K_{X\times Y} \cong \pi_1^*K_X$. Now the pullback of a nef line bundle is again nef, see Proposition 1.8 (i) of Compact Complex Manifolds with Numerically Effective Tangent Bundles by Demailly, Peternell, and Schneider. So one can construct many examples by choosing $X$ with $K_X$ nef and $Y$ non-Kähler with $K_Y$ trivial.
Example: Let $X$ be a curve of genus $g > 1$ and $Y$ be a primary Kodaira surface. Then $X\times Y$ is a non-Kähler threefold with $K_{X\times Y}$ nef. Note that $X\times Y$ is also not Moishezon as it contains $Y$ as a complex submanifold and $Y$ is not Moishezon.
For more examples of non-Kähler manifolds with $K_Y$ trivial, see the introduction of Non-Kähler Calabi-Yau Manifolds by Tosatti. As is pointed out in Proposition 1.1, if $K_Y$ is trivial (or even just torsion), then it admits a metric $h$ with curvature $\Theta_h = 0$ and hence $K_Y$ is nef, so we don't even need to take a product with $X$ in the above construction.