Cut a cube into 5 3D objects with 6 edges each.
(I assume there is only one solution): only 4 of the objects are pyramids, ie tetrahedrons with one vertex containing only right angles
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Draw a diagonal line on one of the faces and the orthogonal line on the opposite face. Join the ends of those lines together. With those as the cut points you get four corners of the cube (each with 4 sides) and the remainder is a tetrahedron with 6 edges.
The above is now correct given the update made by the OP.
Here's a rendering of the dissection done in TinkerCad:
This is a link to the Tinkercad object:
If you go there you should be able to spin it around and stuff.