As usual I have trouble imagining how rule changes will affect things. Just some initial thoughts:
In general it would be hard (or at least harder) to get pieces near the enemy king during the middlegame, because potentially you'd have to sacrifice them if the king moved into their line of fire. Particular the bishops during the early part of the game would be restricted.
And you might think, well then attacks might be more common... but an attack where you're forced to play checks would (usually) end almost immediately because the king would draw them in to be slaughtered.
Actually the king in general would a quite a lot stronger. It may even be an important attacking-type piece in the middle game.
Just as a very simple example
So I was thinking earlier today about nothing. Then I started thinking what would be the simplest rule change that would have the most profound effect on the gameplay. Then I wondered what chess would be like if on your turn if you had a move that could check the opponents King, you were required to make it. If more than one is available the choice obviously goes the the mover. Similar to in checkers (sorry for the cross reference) where one must jump if possible.
Additionally, but not essentially, if one needed to be reminded or shown, it went to the one being placed in check as a "check" (your opponent had a move to check your King but did something else instead and you had to point it out.) After 4 "checks" the forgetful one has a pawn of his opponent's choice removed from his army. Or the one collecting the "checks" can use one to NOT have to move to place the other's King in check, thereby avoiding a trap or bad position. This would be enforceable in live online as well as the computer knows you made an illegal move. This rule could easily translate to 960 or other variants as well.
I'm sure the purists will not be pleased but it would make it different at least and could be fun for beerchess. I dub thee CHECKCHESS.
thoughts?