Are there examples of nonKahler complex manifolds with holomorphically trivial canonical bundle?
Yes, you might look at the following paper by J. Fine and D. Panov: http://arxiv.org/abs/0905.3237
This is covered in Andrei Halanay's answer, but it's worth mentioning the simplest examples, which are primary Kodaira surfaces. For the simplest of these:
Take C^2 and quotient by the group generated by these a_k:
a_1 : z > z + 1
a_2 : z > z + i
a_3 : w > w + z + 1
a_4 : w > w  iz + i
(I think this is it.)
The quotient group is nonabelian. Here z is the fiber and w the base.

$\begingroup$ Let me add that it's obvious that dz dw is preserved in the quotienting (CY). Also, b_1 = 3, meaning it's nonKahler. Here is a link to Kodaira's paper: jstor.org/pss/2373157 $\endgroup$ – Eric Zaslow Nov 19 '10 at 12:59
There is a reasonably extensive literature on nonKahler CalabiYau threefolds. They are of interest in string theory; see for example http://xxx.lanl.gov/abs/hepth/0301161, as well as http://xxx.lanl.gov/abs/0809.4748 for an analogue of Calabi's conjecture in this context.
Any nontrivial principal elliptic bundle $\pi:X \to B$ over a CalabiYau basis is nonKaehler but has trivial canonical bundle (because $\mathcal{K}_X \simeq \pi^*\mathcal{K}_B$).

$\begingroup$ Nice example! How do you see that it is nonKahler? Do you have a reference? $\endgroup$ – JME Jul 30 '11 at 10:53

1$\begingroup$ This is an old result of Andre Blanchard dating back in the '50s. A very thorough discussion may be found in a paper of Thomas Hofer: projecteuclid.org/DPubS/Repository/1.0/… $\endgroup$ – Andrei Halanay Aug 1 '11 at 19:32
One can ask for nonKähler compact manifolds with holomorphically trivial tangent bundle, and still get many examples. By a result of Wang ( Proc. AMS 5 ), these are quotients of a complex Lie group $G$ by a discrete subgroup $\Gamma$.
If the quotient is compact Kähler then $G$ must be abelian. Indeed, every vector subspace of the Lie algebra of $G$ gives rise to a holomorphic differential form on $G/\Gamma$. If $G$ is not abelian then one can choose a subspace not closed under the Lie bracket. The corresponding differential form is clearly not closed, what cannot happen in compact Kähler manifolds.
For a thorough study of examples of this kind see this book by Winkelmann.
More examples are given by nilmanifolds.
Barberis, Dotti and Verbitsky proved in Theorem 2.7 in http://www.ams.org/mathscinetgetitem?mr=2496748 that nilmanifolds endowed with invariant complex structures have trivial canonical bundle. See also Cavalcanti and Gualtieri's Theorem 3.1 in http://www.ams.org/mathscinetgetitem?mr=2131642
On the other hand, nontori nilmanifolds never admit a Kaehler structure, because of Benson and Gordon, http://www.ams.org/mathscinetgetitem?mr=976592 , or Hasegawa, http://www.ams.org/mathscinetgetitem?mr=946638 , or ...