Cubical blocks with the same letter on all sides would almost work. Except that it wouldn't allow words to be build sideways from an upper level without other blocks acting as supports.
There are in fact, a lot of other games from the past that have been referred to as "three-dimensional Scrabble" or something similar, but they never seem to be successful, truly three-dimensional implementations.
Upwords
This patent from 1997 is for methods of making a three-dimensional Scrabble game.
A figure from the patent documents illustrates what I would consider closer to a true 3-D word game:


Another patent was quite recently applied for for a three-dimensional word game. The inventor is listed as Joseph Elie Tefaye with Games R Us of Australia.
This figure shows the essence of the idea:

The design is reminiscent of the two-dimensional Typ-Dom, a predecessor of Scrabble invented in Austria in 1936. In Typ-Dom, the letter tiles are shaped

The extension to making Typ-Dom into a three-dimensional word game just required two main ideas: 1) Using a base to build the three-dimensional word grid from, and 2) modifying the pieces so that one can link up with another at a 90 degree angle. It looks like this might work out nicely.
But if you can't stand waiting for someone else to manufacture the perfect 3-D word game, you could always make your own, Star-Trek-style:
