We consider the problem of constructing Latin cubes subject to the condition that some symbols may not appear in certain cells. We prove that there is a constant $γ> 0$ such that if $n=2t$ and $A$ is a $3$-dimensional $n\times n\times n$ array where every cell contains at most $γn$ symbols, and every symbol occurs at most $γn$ times in every line of $A$, then $A$ is {\em avoidable}; that is, there is a Latin cube $L$ of order $n$ such that for every $1\leq i,j,k\leq n$, the symbol in position $(i,j,k)$ of $L$ does not appear in the corresponding cell of $A$.