At times I sound like a broken record but I really can't emphasize enough how fixed-limit poker (the way most mixed-games variants are played) is a fundamentally different beast than pot-limit and no-limit poker. As a result, many mixed-games players lack certain basic fixed-limit skills that have no use in the pot-limit and no-limit worlds. A striking example of this is knowing when to bet-fold a river rather than check-call it.
Imagine this hold'em situation. You flop the nut straight in a three-handed pot. You're the first player to act and lead out with a bet. One of your opponents calls before the other raises. You call the raise, as does the first opponent. The turn pairs the lowest flop card. Action checks to the player in late position. You follow his bet with a check-raise. That's enough to drive out the third player, leaving you heads-up with the late position player for a river that pairs the middle card on the flop and double-pairs the board.
In this situation, many players in your shoes — with the flopped nuts on a board that has runner-runner double-paired — will "freeze up" and check the action. This has two very unfortunate consequences. First, it allows your opponent to check behind with certain hands (mainly, hands that flopped top pair, whether or not it was with a draw) that might otherwise call a single bet on the river. Secondly, it puts the decision squarely on your shoulders if your opponent bets. Is he just betting top pair? Did he make a full house? Does he have a busted draw? Without an excellent read on your opponent, you never really know and thus you're forced to call down. Most of the time, he will show you a full house — unless he's a total LAG he is unlikely to bet top pair in this spot. The times he has top pair, he checks behind.
Consider, instead of check-calling the river, what might happen if you bet-fold it. Given the strength you have shown so far in the hand, it is very unlikely that your opponent can bluff-raise this river. Thus if your opponent raises, you can safely throw your hand away. On the other hand, he may talk himself into making a "crying call" on the basis of pot size, your bluffing potential, and the current phase of the moon.
When you check-call, you gain nothing if you have the best hand (almost all of the time) and lose one bet when you're behind. If you bet-fold, you gain one bet a certain percentage of the time that your opponent talks himself into calling, and lose one bet when his river raise indicates you're hand has been overtaken. The case is clear, then, that bet-folding is the bottom line maximizer.
The problem that many players have with bet-folding is that it requires a certain level of discipline. Fixed-limit poker is all about maximizing the amount of money that goes in while you're ahead and minimizing the amount that goes in while you're behind. Too many people will bet-call when raised on this river, even though there's virtually no chance their hand is best. They feel like they have made it this far in the hand, and it's only one more bet to get to showdown with more than ten bets already in the pot.
The showdown is not what matters in any form of poker, though. All that matters is making proper decisions to maximize your bottom line. In this case, folding to the raise is a simple matter of discipline. Calling a raise on the river in this spot is futile. No matter how great your starting hand might have been, it is now second-best. Throw it away!
Bet-folding the river is not always the right play in every river situation. For one thing, if you do it every time, you open yourself up to repeated bluff-raises by sharp-eyed opponents. For another thing, it's a tool to be used only under the right conditions. Generally, those conditions are: (1) you have a strong hand that you'd like to get value out of; (2) the other player is at least somewhat likely to call with a second-best hand, given your perception of that player and the action in the hand; and (3) the other player is unlikely to bluff-raise, allowing you to deduce from a river raise that your hand is second-best.
Big-bet poker players would have a completely different view of this hand. It's perfectly reasonable for them to assume that a raise on the river could be an attempt by their opponent to scare them out of the pot. That's what making pressure bets in big-bet poker is all about. In fixed-limit poker, on the other hand, because of the imposed betting structure, bluffing opportunities take different forms. They rarely arise on this kind of board. Knowing this, and using it to your advantage to win an extra bet or save an extra bet, is what fixed-limit poker is all about.