Let $M$ be a 3-manifold fibering over $S^1$, so there exists a fibration $\Sigma \to M \to S^1$. 
Then $M\times M$ will admit a symplectic structure.  

There is a symplectic structure on $M\times S^1$, associated to the fibration $\Sigma \to M\times S^1 \to S^1\times S^1=T^2$ which is trivial in the second factor, by a [result of Thurston][1] (the [converse also holds][2]).  

Similarly, there is a fibration $\Sigma \to M\times M \to S^1\times M$ which is trivial in the second factor. Hence by the result in the fourth paragraph of Thurston's paper, $M\times M$ admits a symplectic structure. 

I think these manifolds cannot be Kähler usually. See the main result of [Biswas-Mj-Seshadri][3].


  [1]: https://www.jstor.org/stable/2041749
  [2]: http://doi.org/10.4007/annals.2011.173.3.8
  [3]: https://arxiv.org/abs/1101.1162