Are Calabi-Yau manifolds in dimension >= 3 algebraic?

It depends a little bit on your definition of CY. If you're using a good one, it will imply that the Hodge numbers $h^{0,p} = 0$ for $p neq 0,d$ (see, for example, Prop. 5.3 of Joyce's http://arxiv.org/abs/math/0108088). This implies that $H^2(X) cong H^{1,1}(X)$. Since the Kaehler cone is an open set in $H^{1,1}(X)$, it contains an rational class, and we can scale that to be an integral class. So, by Kodaira and Chow, we're done.