Scouring through Wikipedia, I've found the following analogs to platonic solids that are composed of irregular faces.
Cube = Trigonal Trapezohedron
Dodecahedron = Tetartoid
Tetrahedron = Disphenoid
I couldn't find the analogs for the Octahedron and Icosahedron. Does that mean they don't exist? How do I prove either way?
EDIT: My bad, Octahedron exists as well. That leaves Icosahedron.
Best Answer
Why, of course it can be done. Start with the net, assign the lengths so that all triangles will be equal and scalene:
Then fold it and glue everything together.
True, the figure looks god-awfully ugly. This is a consequence of the fact that it can't be made face-transitive, unlike your three examples. (To make 20 faces symmetrically equivalent, we'd have to keep at least one 5-fold symmetry axis, which would keep passing through a vertex, which would make its five edges equal, which in turn doesn't sit well with the triangles being scalene.)
If you agree on isosceles faces, then you may keep a 5-fold axis and arrive at Aretino's example.
I suspect there are multiple ways to connect the scalene triangles (this is the case with octahedron, after all), but this post is getting too long, so let's call it a day.