Since this question remained without answers even after a bounty, I thought it might be time to ask it here.

For which pyramid can you compute the volume from simple cut-and-glue processes? The Dehn invariant naturally gives the answer, but I failed to turn this in an algorithm. Here are the pyramids, I know of, whose volume is computable by elementary operations:

• take a cube and divide into six pyramids from its center (or three pyramids given the 120° symmetry along a diagonal). These pyramids can be subdivided and glued into further pyramids, but there are not so many possibilities.

• take a trigonal trapezohedron whose faces are all rhombic. This is not a right prism, but oblique prisms can also be cut-and-glued to compute their volume. Because of his symmetry group you can cut it into three pyramidal pieces (oblique pyramids). Since the small angle of the rhombus can be $\in ]0, \pi/2[$ (at $\pi/2$ it's just a cube), this gives an infinite family of pyramids. (These oblique pyramids have a symmetry: you can cut them further in half.)

But the that's all I could find. Are there any other? and if so, what is the cut-and-glue process?

Some background:

• The volume of pyramids is discussed here in Proposition 3 to 5 of Euclid's book

• The fact there was no "simple" proof irritated Gauß and later Hilbert. It was purportedly the inspiration for his 3rd Problem, which was solved by Dehn (using his invariant)