smc

The cut $x=y$ separates the cube into points with $x<y$ and points with $x>y$, and analogous results apply for the other cuts. Thus, which piece a particular point is in depends only on the relative sizes of its coordinates $x$, $y$, and $z$ - for example, all points with the ordering $x<y<z$ are in the same piece. Thus, as there are $3!=6$ possible orderings, there are $6$ pieces, which is answer $\boxed{6}$.