Dice are awesome, and us gamers are more than familiar with the great variety out there. Most roleplaying games use them one way or another because they're convenient randomisers that prevent games from becoming predictable.
However, sometimes having an alternative resolution mechanic can change the feel of a game and keep things exciting. Dice are great, but they're by no means the be-all and end-all to the options designers have for implementing mechanics. Here are five options for non-dice based RPGs:
There's a lot to be said for the humble domino. Like dice, they allow for numbered resolutions, which can make it easy to implement them into stat-heavy games that require hit points etc. There's also a visual element to having dominoes in a game, as the growing chain of bones denotes progression in a session, whether that's through combat or an action resolution. Imagine a combat where each player has a number of dominoes they can use and each action they do requires them to put down a domino. If they can't go then they have to sacrifice hit points in order to take another domino, taking damage equal to the total of the current ends of the chain.
2. Playing cards
This one's easy enough and is already a mechanic in a few games, such as Deadlands. There are a bunch of ways you can decide how actions are resolved, from playing a few hands of poker (great for Wild West games) to a simple 'higher card wins' mechanic like in War. Like dominoes, you can even make chains with cards and offer a similar hit resolution as above.
A mechanic used by Dread, a horror RPG by The Impossible Dream. It's simple but very effective and leads to some great story-driven games. If you want to perform an action, such as when you're fighting, you remove a piece from the tower. Obviously the more actions there are, the riskier things become until someone sends it crashing down. If this happens they become a 'dead man walking' where if they aren't killed straight away, the next time they would be in danger for their life, they would automatically die. This leads to some great tension - all because of a classic kids' game.
4. Beer pong
Well, not exactly. If you've ever played beer pong as a college deviant, you will know that each side has a row of cups filled with beer and each teams' goal is to bounce or throw a ping pong ball into one of their opponent's cups to force them to down their beer. While drinking isn't hugely practical when it comes to gaming (unless you're playing Drinking Quest) you could always set up a space with empty cups, each cup representing a certain value, whether that be the number of hits inflicted or a representation of how well someone pulls off an action. Skill-based mechanics are quite rare, so this could be very interesting, if a little non-immersive.
Similar to the Jenga resolution, Kerplunk would have players remove a number of sticks depending on the type of action they want to do and the difficulty the GM sets. For example, to jump onto a moving train from a bridge without falling to your death may require four sticks to be removed. The number of marbles falling could denote the hits taken or how the story swings. If all the marbles fall then that's very bad news.