6x6x6 cube with smooth plates on bottom?
While there could be various approaches building studless the easiest one would be with Tile, Modified 2 x 2 Inverted. These modified tiles have studs on one side, which fit under bricks and plates, and smooth surface on the other side.



Here is a prototype which features equal 6-stud (or 30 modules) long sides. Parts can be changed upon availability, but it gives the idea how high you need to go to reach 6-stud height. Red parts are modified tiles mentioned above.