The T-tetromino is a T-shaped figure made of four unit squares. An $m\times n$ rectangle can be tiled by T-tetrominoes if and only if both $m$ and $n$ are multiples of 4. This was proved in a 1965 paper by D.W.Walkup, and the proof was "hands on".

Some "algebraic" tricks like colouring or tiling groups can prove that $mn$ must be a multiple of 8, but they do not seem to rule out the cases like $99\times 200$ and $100\times 102$.

I wonder whether a better proof of D.W.Walkup's theorem is known today. By "better" I mean applicable to non-rectangular regions as well. For example, is there a way to determine what 6-gons (8-gons, ...) admit tiling by T-tetrominoes?