It is convenient to place the image of the tetrahedron with vertices at every other vertex of a cube. I am imagining the cube of side 10 units, whatever they might be.
The seven sides of one of the six pieces, the one going between $(0,0,10)$ and $(10,10,10)$ are given by:
$$ z=8 $$
Two long trapezoids:
$$ x-y+z = 10 $$
$$ -x+y+z = 10 $$
Four triangles, invisible after final assembly:
$$ x = z $$
$$ y = z $$
$$ y+z = 10 $$
$$ x+z = 10 $$
The edges bounding the polygons, if made from thin cardboard as I did as a child, come from pairs of intersecting planes; the vertices from triples of intersecting planes.
I made jpegs of these, if it lets me load them... you can print them out, cut out around the outline, make folds where needed, then use some tape to place the two figures together. Hmmmm. The paper is won't stay flat, very hard to tape together. Better to glue these to some thin cardboard, then cut out those pieces, then put together. Still, I can tell already from the paper version that i got it right.
Version by user Nominal Animal:
Alright, did with cardboard, then lightly scored some of the lines for folding, rather than cutting into all separate pieces. Came out well. I do have a digital camera, so you can get some idea.
Someday in the far future, cryptoarchaeologists will happen upon this site, see that I tried to convince the OP that he ought to build his shape in something easy first, as it has seven sides, and they will say, "It was good"