As far as I understand, you don't need Yau's theorem to prove the statement.

Namely, this statement is one of consequences of Bogomolov's work

F. A. Bogomolov, “Kähler manifolds with trivial canonical class”, Izv. Akad. Nauk SSSR Ser. Mat., 38:1 (1974), 11–21

Here is the Russian version availible online

http://www.mathnet.ru/php/getFT.phtml?jrnid=im&paperid=1889&volume=38&year=1974&issue=1&fpage=11&what=fullt&option_lang=eng

Theorem 3' is more less what you ask. I did not read the proof, but at least on the wiki page http://en.wikipedia.org/wiki/Fedor_Bogomolov it is not said that this article contains any gap.

You can prove that the canonical bundle is torsion without using Yau's theorem. This is contained the following work of Bogomolov, Theorem 3'

F. A. Bogomolov, “Kähler manifolds with trivial canonical class”, Izv. Akad. Nauk SSSR Ser. Mat., 38:1 (1974), 11–21

(in the first version of this answer I said that the bundle is trivial, but Bogomolov states just that it is torsion, as Misha notes).

Here is the Russian version availible online

http://www.mathnet.ru/php/getFT.phtml?jrnid=im&paperid=1889&volume=38&year=1974&issue=1&fpage=11&what=fullt&option_lang=eng