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 gamma amp;gt; 0 such that if n = 2(t) and A is a 3-dimensional n x n x n array where every cell contains at most gamma n symbols, and every symbol occurs at most gamma n times in every line of A, then A is avoidable; that is, there is a Latin cube L of order n such that for every 1 amp;lt;= i, j, k amp;lt;= n, the symbol in position (i, j, k) of L does not appear in the corresponding cell of A.