I think it depends what you mean by higher-dimensional go (and also what “works very well” means). There is definitely something special about the 2D grid, but there are some boards embed in higher dimensions which still use the 2D grid. Examples: toroidal go, Klein bottle
There are also 3D variants that are not based on embeddings of the 2D grid that appear to be playable. Check out the diamond lattice. Diamond Go. Haven’t played it myself so can’t say if it “works”, but at least worthy of exploration.