Here's a technique that is sort of cheating:

Start with a cube, sized as 3x3x3 of the final-cube size. Make two parallel cuts. Now you have three pieces, each 3x3x1 of the final-cube size.

Line those three pieces up and make two more cuts. Now you have nine pieces, each 3x1x1 of the final-cube size.

Now line up those nine pieces in a flat row. This is easiest to envision as occurring on the table of a bandsaw. The nine pieces will be cut sequentially, not three at a time. As each piece is cut, what results is one 2x1x1 piece, and one 1x1x1 piece. As it's generated, immediately move each 2x1x1 piece to the end of the feed row, aligned so that its midpoint will be passed through by the blade.

After the nine 3x1x1 pieces are done being cut, the blade begins to cut through the nine 2x1x1 pieces...all in one feed process, sort of.

When you're done, you've made five cuts, and you have 27 1x1x1 cubes.