To begin with I would like to point out that this article is solely based on limit holdem strategy.
Many players, as a matter of routine, will bet out on the flop with a strong hand but always check the turn. They do this because they don't want to get raised by an opponent on the turn or the river (when the bets are doubled). This is a pattern you often come across in low limit cash games. On the higher levels you will be confronted with this play a lot less frequently. However, sometimes the medium- and high-stakes tables attract players with this weakness, so it's worth analyzing this play and coming up with ways to exploit it.
Observe your opponent
Let us assume you are paying close attention to your opponent and his betting pattern: This opponent gets A-10 and sees a flop A-8-4. Your opponent bets on the flop and two players call. The turn shows a 3 and your opponent checks. If someone behind him bets, he will simply call the bet. On the river he will check/call as well. Based on this observation it looks like this opponent seems to be a weak or weak-tight player.
Keep watching this player closely and see how he plays during a similar situation in a alter hand. If his betting pattern remains the same, the time has come to figure out a strategy to exploit this weak play.
Think thoroughly about the motivation fort his type of play. If he is a weak and a tight player, he won't bet the flop and check the turn because he has simply given up the chance to bluff or because he thinks he got outdrawn on the turn. He is simply scared to get raised on the turn or the river when the bets have doubled. For that reason he will just check the hand with the intention of calling you down to the river. Most of the time its just that simple.
How to play against this sort of opponent
There are two big adjustments to your game that you can undertake, based on the observation of your opponent.
First of all, you can call with marginal hands before and on the flop when you are in late position and your opponent is about2 to 3 places to your right. For example, if you have 6-7 off suit in late position, you should probably not play this hand unless a big multi-way pot is forming. If it is a medium sized pot (about 3 to 4 opponents) and this specific opponent calls preflop, then you might want to join in. If you manage to hit any sort of draw on the flop, then there's a big chance that this opponent will give you a free card on the turn. This is why you can play more drawing hands against these opponents, even if your draws aren't even that strong.
For example, imagine there are five callers (including yourself and this opponent) before the flop and you hold 6-7 off-suit. The flop shows K-8-4 and it's checked to this opponent. He bets. Now you have to call 1 small bet in order to win the 6 bets in the pot, and some of the layers behind you might still call as well. But let's just say you are getting 6:1 pot odds on the flop. Should you call?
The possibility of hitting a 5 on the turn or the river is about 5:1. Then again, if you don't hit your five on the turn you might have to fold your hand to a bet. Your odds of hitting a 5 on the turn or the river are 5:1, and because you are likely to get to see both the turn and the river card, you can take the 5:1 odds because you get pot odds of 6:1, and this is without taking the implied odds into consideration and the chance that players in early position will call and therefore increase the pot odds even more.
This approach comes with three warnings, based on the same reasoning. First of all, if this certain opponent bets again on the turn, you will have to fold our hand. If it seems like this opponent is becoming unpredictable in his actions, you will have to give up this strategy.
Second of all, if a second opponent at your table has analyzed this players betting pattern just like you, and start ruining your strategy by firing out check-raises on the flop or bets out of position on the turn, then stay out of his way. Wait for the right moment. This second player could make your target even weaker, which means more free cards and as a result, more pots.
And thirdly, if your opponent is in early position, this play won't be as effective. If he bets the flop and checks the turn, then players in middle and late position between you and your opponent will often bet the turn, which ruins your free card strategy.
Apart from the free card strategy against this type of player, you should never try and bluff or semi-bluff this player on the turn. In my experience I have won more money by accepting the free river card than I have by semi-bluffing. This is because of the motivation behind this kind of behavior. He has a good hand but checks the turn to avoid being raised when the bets have doubled. That is why he has decided to call you up to the showdown and he probably has a hand good enough to do so. Don't bluff a caller!
When to use this approach
Finally, you could use the "bet the flop, check the turn" strategy yourself when your playing heads-up against a maniac with a pretty good hand. Imagine you have A-Q. The maniac raises, you call and you're heads-up. The flop shows A-8-6. If you bet here (which I think you have to do), he will probably raise, in which case I would check the turn and just call down your opponent on the turn and the river. This way you avoid losing a lot of money when you're behind and you avoid getting pushed off the best hand.