Here’s a thread about it on ENWorld with some examples: http://www.enworld.org/forum/general-rpg-discussion/194714-dread-jenga-beat-up-my-dice-my-results-indie-horror-rpg.html

I do employ alternative mechanisms and they can be a strong plus.
First a few general principles. I feel that alternatives are good if not overused : a change of pace, a different kind of challenge. I feel they should be simple (touch-test : how fast can you explain it? In less than 60 seconds?)

Puzzles are good for unusual situations IF the main PC being put to the test is played be someone capable of solving the puzzle/test you put before them. Also, the other players should be able to contribute. This last criteria strongly implies that one player shouldn’t be excluded because he’s consistantly lousy at that type of puzzle/game. A team effort can overcome this somewhat, but be ready to dump your wonderful challenge if three-quarters of your players exclaim about how unfair this is on them and how they’ve suddenly been condemned to failure….
So I keep this sort of thing for occasional challenges not as a replacement for the main success-determination mechanism.

A few alternatives I used /run across:
“How to assist a ritual”. 5 black and 5 white balls in a bag plus one white ball per person helping pray/cast the ritual – more if they are powerful/strongly motivated (GM call based on their role-playing). At the conclusion of the ritual draw a white ball to succeed (a black ball=failure). Good for ressurection of fallen comrades and BIG ceremonial magics.

“Logic Puzzle”*. Paper grid of numbers 1 to 100, and clues such as “It’s not a prime number”, “It’s not divisible by seven”, “It’s not in the third column” to eliminate all but one of the numbers. Tell the GM the number you’re picking to succeed. All players can chip in comments on this one.
“Logic Puzzle II*”
Several clues about a situation have to be cross-referenced
(e.g. “Mary lives on the left hand side of the street, her family name does not have an odd number of letters”; “Mr Baker does have a daughter named Jane, but his house doesn’t have a green door”, “All the doors on the houses with an odd number on them are white”)
All converge to eliminate all-but-one answer to the question “Which parent-and-child lives at number 37 and what is the colour of their front door?”. Good for murder mysteries document-research.

*=There are entire magazines of these kinds of puzzles and I’m sure there must be internet sites with free examples.

“Pentominoes” : The GM says “Assemble a Rectangle 5 by 12 to succeed” (N.B. Only works a few times before the players memorise solutions…. and the 3×20 rectangle is a monster!)or “Finish this rectangle to succeed”. Also called “polyominoes” in Arthur C Clarks book “Imperial Earth”. Can be constructed from cardboard.

Manual dexterity tests (NOT for klutzes!)
Those little frames that have sliding-tiles that you must rearrange into a coherent picture (possibly within a GM-imposed time limit. Good for thieves who need to defeat a lock/trap within a set time-limit. Ditto those little steel ball-bearings-balancing-test (inside hollow perspex cubes). They can be a nasty challenge. Hard to match to a players skill, too. But there is potentially a definite “Aha!” moment with a cool visual proof of success at the end….
It’s a shame these aren’t infinitely reusable and modulable like dice….

Recap:
A change of pace, a different kind of challenge. Good if not over-used, if rapidly-explainable.
Can help players make fond memories, especially during a team effort. Visual supports help this.
May require careful matching to players skill levels, and/or suddenly dumping it to fall back on dice-rolling in the face of potential mass player reticence.
Examples of puzzles can be found on the internet.