Here are the changes in the steps needed to make the other platonic solids:

cube

• gather materials and tools
  • 27 cards, 24 for the cardahedron
• cut template card
  • use 45° and 45°
  • guide
• join cards
  • 4 cards join at each corner, and 4 join at the center of each face

octahedron

• gather materials and tools
  • 27 cards, 24 for the cardahedron
• join cards
  • 4 cards join at each corner with the 30° slot, and 3 join at the center of each face with the 60° slot

dodecahedron

• gather materials and tools
  • 63 cards, 60 for the cardahedron
• cut template card
  • use 54° and 36°
  • guide
• join cards
  • 3 cards join at each corner with the 54° slot, and 5 join at the center of each face with the 36° slot

icosahedron

• gather materials and tools
  • 63 cards, 60 for the cardahedron
• join cards
  • 5 cards join at each corner with the 30° slot, and 3 join at the center of each face with the 60° slot

It is also possible to make stellated solids with this method... and changing to an edge-oriented technique (where a folded card forms each edge) allows lots of irregular solids too...