Some fairly obvious cheating

The Scrabble world has been rocked by a cheating scandal at the US National Championship. An unnamed junior player was expelled from the tournament after he was caught hiding tiles under the table. Apparently he was 'palming' the blank tiles, rather than dropping them into the tile bag before the start of the game. There is a pretty big article about it here, and plenty more if you search the web.
Of course chess has its own recent cheating scandal (involving computer assistance), but it is rare that cheating in chess involves changing the 'state' of the game.
There are a couple of times when I have seen or heard it happen over the last 25 years. I know of at least one player who removed a players piece from the board when they were absent, but this was noticed very quickly and appropriate action was taken. I've also seen an incident where a player was accused of doing this, but it turns out the opponent had forgotten he had blundered a piece 10 moves earlier.
A more subtle attempt of cheating was to adjust the times on the clock. This only worked with the old analog clocks, as you could sneak your hand behind the clock and operate the winder. Again I am aware of at least one incident involving this method, and the player concerned was caught in the act by a spectator.
Of course at blitz chess there are still a number of methods, which in a sense are so common that the term 'cheating' often isn't applied. The 'disappearing queen' is still an old favourite, where in an ending, the queen you wish to promote to has 'fallen' off the board. In this case the few seconds needed to either find it or work out what is going on are often crucial, but as always, your opponent is most apologetic for the 'accident' after you have lost the game.