I am trying to gather a list of all known symplectic manifolds which don't have Kahler structure. If you know any please add to the list and give references for it.
Please avoid giving repetitive examples.
Thanks.
I am trying to gather a list of all known symplectic manifolds which don't have Kahler structure. If you know any please add to the list and give references for it. Please avoid giving repetitive examples. Thanks. 


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. 


There are lots of simplyconnected fourdimensional examples in Gompf's symplectic sum paper paper which are shown to be nonKahler by virtue of violating the Noether inequality (see Theorem 6.2). There are also higherdimensional examples in the last section of Gompf's paper, including infinitely many simplyconnected nonKahler ones in any even dimension at least 8. Some of these are obtained directly by symplectic sum, and others by taking fourdimensional examples and embedding them in CP^n and blowing up, as suggested by Tim and eigenbunnyin these cases Gompf uses the Hard Lefschetz theorem to prove that the result isn't Kahler. The standard symplectic surgery operations in fourdimensions (symplectic sum, Luttinger surgery, rational blowdown...) should generally be expected to produce nonKahler manifolds more often than not, though it's not always feasible to see that the result isn't Kahlerthe standard ways of doing so are by the parity of $b_1$ or by the Noether inequality. In particular this paper of FintushelParkStern gives another large collection of simply connected examples violating the Noether inequality. One can also show that a symplectic manifold is not Kahler by showing that it is not formal (in the sense of rational homotopy theory). Often nonformal examples also don't satisfy hard Lefschetz (so could instead just be shown to be nonKahler by that criterion), but there is a nonformal example of Cavalcanti which does satisfy hard Lefschetz. 


I would recommend the TralleOprea book, Symplectic manifolds with no Kähler structure. 


The original example of a torus bundle over a torus which is symplectic but admits no Kahler structure is due to Thurston. 


Edited: Corrected after Eugene Lerman's comment of Sep 22, 2012, which also implies this is not an answer to the question (edits are in italic). In [Invent. Math. 131 (1998), no. 2, 311–319] Chris Woodward gives examples of multiplicityfree compact Hamiltonian manifolds that are not compatibly Kähler. This proved that Delzant's result that all compact multiplicityfree torus actions are compatibly Kähler [Bul. Soc. math. France 116, 315–339 (1988)] does not extend to the nonabelian case. Woodward uses a result from Susan Tolman's paper "Examples of non Kähler Hamiltonian torus actions" [Invent. Math. 131 (1998), no. 2, 299–310]. So, backtoback papers with examples of symplectic manifolds with a lot of symmetry that are not compatibly Kähler. 

