The free-showdown play used to be a very popular play in limit hold 'em. The play is as follows: when heads-up and in position with a hand that has some showdown value you raise the turn and check the river unimproved.1 This play costs the same as calling on two streets, but it gives you a chance of winning the pot immediately, can prevent opponents from outdrawing you, and gives you the option of extracting further value by betting the river if your hand improves.
These advantages lead to the play being very popular for a while, particularly amongst highly aggressive lower stakes online professionals. However, the play also has some serious disadvantages, and this is why it is used much less nowadays.
For starters, the play creates problems whenever your opponent shows more aggression, either by reraising the turn or by betting out on the river. Either of these occurrences can lead to difficult decisions, as calling down wastes a number of bets, but folding always leaves the potential that you may have been bluffed out of the hand. Calling down avoids both of these problems, and by calling the turn you can still often get a raise in on the river if your hand does improve. Calling down also induces bluffs from worse hands, allowing hands that would've folded to a turn raise to lose another bet by bluffing the river.
There is, however, another overriding factor why the play has fallen out of favour. The free-showdown play essentially turns a hand that was strong enough to call down, a bluff-catcher, into a bluff. When attempting to implement a balanced overall strategy, it makes the most sense to call down with all your positive-expectation bluff-catchers, and then to bluff-raise with the strongest hands you would otherwise fold. Playing this way will extract the greatest amount of value from your entire hand range. The free-showdown play goes against this criterion, and that is ultimately why strong players have largely stopped using it.
Think of the spot in this way: you will want to be raising the turn for value with some of your strongest hands. In addition you'll be raising the turn with some strong draws such as flush draws and open-ended straight draws. These semi-bluffs are in effect part value raise part bluff. Other hands will want to call the turn, either intending to call down unimproved or calling because their equity is higher than the pot odds on offer. You'll then be raising the turn with some weaker bluffs. These bluffs are effectively there to balance your value bets: if you never bluffed then your opponent would be able to read you too easily. Unless you are attempting to exploit an opponent who folds too much, there is a sharp constraint on how often you bluff. You cannot bluff too often or else your opponent will be able to exploit you. The free-showdown play takes a hand which was strong enough to call and turns it into a bluff. However, you would gain more value from your entire range of hands if you instead bluffed with a hand which was too weak to call the turn – as hands that are too weak to call can only continue by bluffing, while free-showdown hands can always call profitably instead.
There is one specific situation where I think the free-showdown play is due for a reprisal. In order to get the same number of bets in, the play's alternative, calling the turn and raising the river if you improve, relies on your opponent frequently following through by betting on the river. However, this might not be the case in instances where the board is incredibly wet. For example, if a fourth connecting card comes on a three-flush or three-straight board, then your opponent will check-and-call with many marginal value hands, such as top pair, which would have bet on a safer river. If this is the case then only raising on the turn will let you get full value from hitting your draw on the river. Here's an example hand:
$10/$20 limit hold 'em – 5 players
Pre Flop: (1.5 SB) Hero is BTN with K 
Q 
1 fold, CO calls, Hero raises, 2 folds, CO calls
Flop: (5.5 SB) 5 J 8 (2 players)
CO bets, Hero raises, CO calls
CO makes an unusual play – betting out into the preflop raiser, usually the sign of an unsophisticated player. I would just call here, because I will be waiting until the turn to raise hands like A J (hoping that another heart doesn't come), and with my flopped flushes too. Since it's a good idea to correlate your semi-bluffs and value raises, calling is the play I'd make here.
Turn: (4.75 BB) J 
(2 players)
CO checks, Hero bets, CO folds
Assume Hero had called the flop and CO continuation bet the turn. I'd make a free-showdown raise here, with the intention to bet the river if a heart, King, or Queen comes; checking otherwise.
The alternative play would be to call the turn and call the river unimproved. This is usually the play I would favour, but if a fourth heart comes on the river your opponent will very often check – allowing you to only get one bet in on that round. The free-showdown raise allows you to get a total of three big bets in after hitting your draw (two on the turn and one on the river).
The play does come with costs. Your opponent could reraise the turn, making your draw more expensive; or he could potentially bluff you off the winning hand. These problems don't exist with calling down, and they are part of the reason why I don't usually advocate the free-showdown play. However, this raise is with a tiny part of your range. I will also be raising the turn with many strong hands which will welcome a reraise. Given that this play is such a small part of my range here, and such a small element in my overall poker strategy, I think the benefits of raising the turn outweigh the costs. Situations like these are the exceptions that prove the rule about not free-showdown raising.
1 One example from Nick Grudzien and Geoff Herzog - Winning In Tough Hold 'em Games: the authors recommend free-showdown raising the turn with 6c6h on a JdTd2h3c board. See pp.181-184.
