a n u
r a s
i u e

And i need to search ani, so, word exists taking an a from second row, n from first one and i from last one.

private static boolean search(String wordToSearch, char[][] matrixCube, int[][] solution, int searchLetterPosition) {
    if (solution[solution.length - 1][0] != -1) {
        printSolution(solution);
        return true;
    }

    char letterToFound = wordToSearch.charAt(searchLetterPosition);
    for (int cube = 0; cube < matrixCube.length; cube++) {
        if (cubeIsFree(solution, cube)) {
            for (int faceInCube = 0; faceInCube < matrixCube[cube].length; faceInCube++) {
                boolean letterFounded = matrixCube[cube][faceInCube] == letterToFound;
                if (letterFounded) {
                    solution[searchLetterPosition][0] = cube;
                    solution[searchLetterPosition][1] = faceInCube;
                    return search(wordToSearch, matrixCube, solution, searchLetterPosition + 1);
                }
            }
        }
    }
    return false;
}

private static boolean cubeIsFree(int[][] solution, int cube) {
    for (int[] aSolution : solution) {
        if (aSolution[0] == cube) {
            return false;
        }
    }
    return true;
}