I think that originally the notion of crepant resolution $f\colon Y\to X$ was given when K_X is Q-Cartier, so that it makes sense to consider discrepancies of exceptional divisors in a resolution. If K_X is not Q-Cartier, it may very well happen that f is small and contracts curves having non-zero intersection with K_Y.

It is easy to see what happens to toric divisors, but it is more difficult to study a general divisor in the linear system $|p^*\mathcal{O}(1,1,1)|$. If $H'_1\subset X'$ is the pull-back of $\{pt\}\times\mathbb{P}^1\times\mathbb{P}^1$, the transform $H_1$ of $H_1'$ in $X$ is the transform of a hyperplane in $\mathbb{P}^5$ containing 2 of the blown-up lines. Similarly you can consider in $X$ the transforms $H_2$ and $H_3$ of hyperplanes in $\mathbb{P}^5$ containing the 2 other pairs of blown-up lines. Then $\mathcal{O}_X(-H_1-H_2-H_3)$ corresponds to $p^*\mathcal{O}(-1,-1,-1)$.