Honestly, this is like asking for a list of all animals that are not dolphins.

Instructions for making your own:

Step 1 - use the symplectic mapping torus for a symplectomorphism of your favorite Kaehler manifold (acting suitably on homology) to produce something whose Betti numbers violate being Kaehler (one odd degree Betti number is odd)

Step 2 - take the manifold from Step 1 and embed it symplectically in projective space, then blow up. That gets rid of the fundamental group, but raises the dimension a lot.

Step 3 - take Donaldson hypersurfaces to get the dimension down again.