The great debate, to dice or not to dice: what is the point? In my opinion, dice are what separate writing a story from playing a game. Both are very enjoyable and worthwhile activities, but if one wishes to engage in a role-playing game dice are a necessity.
Due to my inherent lazy nature, I will be using the term dice as a catch-all for any random system a game uses to determine the outcome of a given action, because that is truly the point of dice: to be random and bring the unexpected into your story. The problems start to come in when you allow the result of the dice to trump the enjoyment of the game. The key is to understand the true function of dice and when and how to use them to enhance your experience, which is not always when the rules tell you to.
Determining direction, not success.
The key thing to remember is that no one controls the dice. Every time you pick them up you are choosing to put the fate of your game in the hands of random chance, which is not a bad thing, it's just not always a good thing. Consider this when designing your encounters. Ask yourself, "How will the game continue if the players fail at the task at hand?" If failing the dice roll would effectively end the session, then you probably shouldn't leave it up to chance.
"So what? Do I let the players succeed at everything just to keep the story moving?"
Not at all, although you can't control the outcome of the dice, you can control the outcome of failure. Encounters shouldn't be pass/fail exams. An encounter is like a fork in the road, passing the test means you get to take the easy path, and failure indicates you will have take the longer potentially more difficult road. Both roads lead to the same location, but one gets you there with less of a cost.
The Joy of Chaos
As the old saying goes, "a plan is just a list of things that never happen," this is what I find enjoyable about gaming. You show up with your prepared story, your preconceptions of how it will unfold, and then expose it to the random chaos of the players and dice. In the end, you wind up with something completely different than what you were expecting, having to react just like the players to the unexpected, trying to keep the ship afloat. Dice allow this chaos to occur even if you know every move your players are going to make. They keep everyone on their toes, and keep the game from being mired down in the monotony of predictable patterns.
Speaking of monotony, it can be difficult at times for a GM to keep coming up with fresh story ideas, this is another point where dice are a handy tool. Random roll tables have gotten a bad rap at times, as many of us remember the days where entire adventures were determined by comparing the roll of the d100 to a series of lists.
As I've matured as a gamer, I began to realize the value of the random roll table is that of inspiration, and in a way, a game in itself. Randomly determining the type of creature, plot, terrain, or even weather conditions can be the spark of inspiration on which your story can grow. The key is to keep what you roll no matter how bizarre -- in fact, the more bizarre, the better. Nothing will get your creative juices flowing like having to figure out why there is a swarm of ice demons running around the inside of an active volcano, or how a dragon ended up in a room with a door too small for him to fit through.
When you run a sandbox game you have to contend with the players getting wild hairs up their asses and running off in random directions, most of which you didn't prepare for. Having the ability to generate some random elements can keep your game from grinding to a halt.
So hopefully, you've stopped worrying and learned to love the dice.