The Chinese and Japanese forms of chess (Xiangqi and Shogi) do not have stalemate.
Chess without check

Check and checkmate are such old rules, that it's just considered the "laws of chess", without the alternatives being considered much anymore (except for the fact that other chess-like games like evert823 mentioned might not have check and stalemate).
If moving in check was allowed, I think there are types of this move:
1) A player moves into check by "mistake". Then the player loses due to a stupid move (a blunder).
2) A player moves into check because he has no other options. Then his king is captured, and loses. In this situation (as in your diagram), the game would have formerly been a "stalemate" and therefore a draw. With the new rule (no checks), stalemates essentially become a win for the player who causes his opponent to be stalemated.
I think this could certainly be a valid and strategic alternative to chess. But I would not prefer it basically because it allows one side to win without a masterful demonstration of checkmating an opponent. One side supposedly rules the board, but any type of capture, artful or clumsy is allowed. Kings are a "royal" piece. If you want to prove you are superior to your opponent, the king gets the priveledge that he must be checkmated. Any other type of "capture" is not worthy of royal piece, and doesn't deserve to be called a "win". So that's why I like "checks" and "checkmates".
I believe every stalemate relies on restriction of the king by hostile pieces or pins (or both).
This is not entirely true, but positions were you are stalemated without this are highly artificial, and never occur in games.
There have been endless discussions on chess.com on whether stalemate should be a win or a draw. Largest impact is that K+P vs K would be won much more often. Next-largest effect probably that K+2N vs K also is a forced win.

I've thought about playing chess with no checks before, allowing loss by blunder, or more like allowing more losses by blunder! Another thing that might change is that you could presumably castle through check, where you're not allowed to at the moment. That rule seems to hinge on some notion of an intermediate state where your king is in danger, but in the normal turn based game there's no actual opportunity for your king to be captured during castling, because the other side doesn't get a turn until you're safely ensconced.
I've thought about playing chess with no checks before, allowing loss by blunder, or more like allowing more losses by blunder! Another thing that might change is that you could presumably castle through check, where you're not allowed to at the moment. That rule seems to hinge on some notion of an intermediate state where your king is in danger, but in the normal turn based game there's no actual opportunity for your king to be captured during castling, because the other side doesn't get a turn until you're safely ensconced.
In this scenario, would your appointment be able to win the game after you castle through check by capturing your king en passant?

Another thing that might change is that you could presumably castle through check, where you're not allowed to at the moment. That rule seems to hinge on some notion of an intermediate state where your king is in danger, but in the normal turn based game there's no actual opportunity for your king to be captured during castling, because the other side doesn't get a turn until you're safely ensconced.
In this scenario, would your appointment be able to win the game after you castle through check by capturing your king en passant?
Castling was originally played in 2 moves, the first being a rook move next to the king, the second being the king's jump over his rook. If we stand true to the origins of castling, the intermediate square occupied by the rook in modern castling is never touched by the king. However, since it used to be done in 2 moves, the king's initial square would remain perilous if we allow king capture. Therefore, I find it most sensible in this case to allow castling "through check", but if one were to castle "out of check", then the opponent should be able to capture the king en passant on its initial square.
The notion of check in chess has always seemed a little arbitrary to me. In effect, the object of the game is to capture the opponent's king, only it never happens due to the priority of check. The game ends when capture becomes inevitable on the following move. The rules are quite clear and consistent, but the idea that you cannot make a move that hangs your king seems like a strange exception: In this one case, you're not allowed to make a move that immediately loses the game. But of course there's no rule against moving into a mate-in-one by force or blunder, hanging your queen, etc.
I started thinking what the game would be like if that one rule were to change. Suppose the object were actually to capture the opponent's king, and there were no such thing as check or checkmate. Making a move that would result in check would not be illegal, just deleterious.
It doesn't seem like it would have much impact on openings, middlegame play or tactics. But I think there would be one enormous difference: No more stalemate. I believe every stalemate relies on restriction of the king by hostile pieces or pins (or both). In the position where the board contains only a white king on h1, a black king on h3 and a black pawn on h2, with white to move, black would win by force. White would have three legal moves: Kxh2, Kg2 and Kh1, all of which would result in capture of the king.
There would still be draws by repetition and perpetual check as well as insufficient material and the 50-move rule. Just no more stalemate.
The so-called draw death of chess, whereby a 12-round championship match can end up with only two wins, could be dramatically affected by a simple, intuitive change to the rules without sacrificing much else in the game.
I doubt I'm the first to have such a thought. Has this been discussed at any length elsewhere?