Yes, there are a few four-dimensional chess variants:
Sphinx Chess, by V.R. Parton, is probably the first, c.1970. Timeline is also described there (it's actually a full 4 spatial dimensions, being 4x4x4x4, plus the "timelines"), as well as my own variant TessChess.
Most 3D chess (at least online) is played by splaying the levels out next to each other, since seeing and reaching between levels may be difficult. Once you get used to that, it's not so unnatural to put the levels in a grid, and you've got four dimensions.
Also, most 3D chess is on a smaller board than 8x8x8 (8x8x3 or 5x5x5 are common), and higher dimensions similarly shrink the edge length. This, together with appropriately extending piece movements, get around NoseKnowsAll's density argument. But, of course they are also right that the game becomes rather different from chess.
V.R. Parton also created a 6-dimensional game, Ecila. I and others have toyed around with even more dimensions, but it obviously becomes more difficult for the "array of array of arrays" board simulation to work out.