Challenge: How To Make 1-inch Wooden Cubes?

I want to make a Soma Cube (a kind of puzzle, which you can read about here) for my kids (and myself).

But I’m having a hard time making accurate wooden cubes from 1-inch x 1-inch wood sticks. I tried using a miter saw but there is too much error. I know it is possible to buy inexpensive wood cubes, but I am interested in finding out how people can make them at home, especially with hand tools.