Truncated octahedron - Cube (11 pieces)
Starting with the truncated octahedron, this dissection can be performed with just four straight cuts.
Rhombic dodecahedron - Cube (11 pieces)
Stellated rhombic dodecahedron - Cube (12 pieces)
Truncated octahedron - Rhombic dodecahedron (24 pieces)
Triangular prism - Cube (10 pieces)
Pentagonal prism - Cube (10 pieces)
Hexagonal prism - Cube (8 pieces)
Starting with the prism, this dissection can be performed with just five straight cuts.
Heptagonal prism - Cube (11 pieces)
Octagonal prism - Cube (10 pieces)
Unfortunately the shaded piece is very thin in the middle, so there is room for improvement here.
Cube - 2 Tetrahedra + Octahedron (15 pieces)
Starting with the cube, this dissection can be performed with just five straight cuts.