Pre-drawing cards or tiles
We've played it slightly different than tttppp; we allow pre-drawing of tiles, as long as the drawing order remains unchanged, and there is no discussion of the tile you have pulled.  

For example, if we have players 1, 2 and 3, and it is player one's turn, player 3 can draw their next tile as long as player 2 has already drawn their next tile.  Yes, tile drawing is random, but we prefer that the tiles are still drawn in order.

The no-discussion portion keeps the spirit of the game I think, because normally you wouldn't have pre-drawn a tile, and you wouldn't have that extra information from the pre-drawn tile.

Pre-drawing makes the game go much faster, since your decision making can be done in advance, as long as a player before you doesn't take the space you wanted, and our groups have no problem with it.