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 multiplicity-free compact Hamiltonian manifolds that are not compatibly Kähler. This proved that Delzant's result that all compact multiplicity-free 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, back-to-back papers with examples of symplectic manifolds with a lot of symmetry that are not compatibly Kähler.