A number of years ago I read about a variant of chess called "One check chess" (or 'first check chess') where the first player to give check wins. Apparently it did not last long as an alternative as Frank Marshall quickly found a forced win for White. Annoyingly a quick search has not turned up the winning moves, although I assume 1.Nc3 (threatening 2.Ne4 or 2.Nd5) seems to be an obvious start.
One possible improvement is 'Safe check chess' where the first player to give a 'safe' check wins. By safe check I mean a check that does not result in the immediate capture of the piece as a legal reply. This would rule out any kamikaze attacks by the knights, although if one side decides to "turtle" it may be difficulty to actually win the game.
A variant which still seems to have legs is "Three check chess". Three checks are required to win the game, and they can happen at any stage (ie non consecutively is fine). Here is an example game where White goes horribly wrong. 1.e4 e6 2.d4 Bb4+ 3.c3 Bxc3+ 4.Nxc3 Qh4 and Black will play a third check with the queen next move. I suspect that this is the kind of chess variant that might be useful to know if you are embarking on a career as a chess hustler, as it is simple enough to entice the unwary, while giving you an edge if you know what you are doing.