The dimensions $h^i(\mathcal O_X)$ of the cohomology groups of $\mathcal O_X$, and thus the Euler characteristic, are birational invariants of proper varieties *in positive characteristic* as well, by a recent work of Andre Chatzistamatiou and Kay Rülling. It is not published yet but a <a href="http://arxiv.org/abs/0911.3599">preprint is available.</a>