To make a three-dimensional honeycomb with the open-cell structure of the diamond crystal lattice, you make many sheet-metal strips of the same width, and you make a stack of two of these layers of strips, running at right angles, either with or without spacing between the strips. This gives you a grille of intersections between the strips. Color these intersections chessboard-style and spot-weld the black ones. Add a third layer at right angles to the second, each of its strips in the same position as a corresponding strip in the first layer (except displaced in Z by twice the sheet metal thickness), and spot-weld the white squares. (This may require a spot-welder with both electrodes on the same side of the workpiece, like those used for welding nickel strips to lithium batteries.) Now at every intersection the second layer is welded to either the first layer or the third layer. Add a fourth layer parallel to the second, and spot-weld it to the third layer at the intersections where the third layer isn’t welded to the second (but the second is welded to the first). And so on.
Once you have added enough metal, pull the layers apart, permanently bending the strips so that each welded intersection becomes a tetrahedral lattice point.
If you have two-dimensional square mesh available for free, you can do this with half as many spot welds. Place a second layer of mesh on top of the first layer, with its X and Y axes parallel, but offset in both X and Y by half a cell, so that each wire in one layer of the mesh crosses a wire in the next layer exactly halfway between the two nearest intersections in each of their respective meshes. Spot-weld all these crossing points; repeat.
This isn’t a very rigid structure, which is why it’s possible to bend it into shape by pulling on it once it’s welded up. If impact energy absorption is the goal, then that’s fine; it should work great for that. However, if higher rigidity is desired, it’s possible to take advantage of metals’ work-hardening tendencies to get it. Say that two intersections are “metamours” if they are directly connected to a single common intersection. The trick is to add additional members to the structure connecting each pair of metamours which get further apart during the pulling process, of which I think each intersection has 8; these extra members are initially not straight, but the initial expansion of the matrix straightens them, which work-hardens them. If the expansion is done rapidly enough, the mass of the centers of these members comes into play, makng them straighter than the same amount of pulling force could have made them under quasistatic conditions.
This may be enough to get an ideal omnitriangulated mesh like an octet truss. There are still 4-cycles in the structure that have no diagonals, one between each pair of metamours in the same layer, but that's true of the standard octet truss as well.