It's a simple concept, really. Just follow this quick and dirty guide.

-You MUST be able to ride a bike with your hands.
I cannot stress this enough. I shudder while thinking of all the poor people who tried to ride their bikes without hands without knowing how to ride it with.
-You must have a fairly old bike.
The reasons for this are twofold. The obvious reason is that falling, something which you are bound to do, will not have as much monetary impact. The second reason is surprising -- it is much easier to ride an old bike without hands than a new one. For some reason the handlebars are stiffer and do not turn as much.
-You must not be afraid to fall.
The path to success is fraught with hardships, young grasshopper.

How to do it
It's rather simple, really. However, there are some things you must keep in mind. For one, get up to a good pace before removing your hands. If you find it hard to keep your balance, the most probable cause is that you simply are not going fast enough. Remember when you first learned to bike? You had to bike quickly to keep your balance. There is some scientific explanation for this.
Secondly, you must conciously keep your balance. For example, if you feel your bike tipping to the left, you must conciously change your centre of mass to the right. You do this unconciously while riding with hands, but you must learn it again from the start now.
And lastly, make sure you are in an open area without pedestrians. I'm sure you can figure this one out for yourself.

Biking without hands is rewarding and useful. For example, now I can drink caffeinated water and read my Calculus textbook at the same time, while biking. My next project is to figure out how to do this safely. However, do not let my isolated incident dissuade you. Go forth and bike!

Adding to Ulumuri's excellent writeup, a few more items seem to be in order:

-The tires matter.
Iffen there every were an important part of a wheeled vehicle, the tire would be it. As Phish sings, "The tires are the part of your car that make contact with the road." Racing treads make riding sans hands far easier. A racing bike, even with its thin contact patch, is more stable than a mountain bike with its asymmetrical knobby bits. A street bike is ideal, with its smooth, wide tires.

-You're riding a gyroscope.
The reason why you are more stable at high speeds is that you have higher rotational momentum in your wheels at high speeds. Just as linear momentum makes it harder to change course, rotational momentum makes it harder to tilt the bike: to fall over, the bike must rotate on the axis parallel to your path. Rotational momentum is dependant on moment of intertia (sort of like the mass of an object when dealing with linear momentum) and rotational velocity (sort of like velocity when dealing with linear momentum). You can increase the rotational velocity by making the wheels spin faster. This involves either large cliffs and furious pedalling or simply riding faster. Ideally, for stability you would ride a bike with wheels simply consisting of large lead disks. This would also give you amazing quadracep.

While fairly easy, never try this on a motorcycle.
Mind you, I'm a wussy, but it really isn't smart. It's easy to be enchanted by a brand-new throttle lock, but one needs to be able to steer, use the front brake, and clutch to operate a motorcyle safely. You would be stable, though. Historical examples of unintentional empirical testing have revealed that the immense rotational momentum of motorcycle wheels at reasonable highway speed can keep a bike up for at least a mile after the rider's unexpected departure.

To correct ulumuri somewhat: the important characteristics of the machine are:
(a) that the frame is "in track" - the front and rear wheels are in the same plane when riding in a straight line. Generally the better quality a bike is (and the less maltreatment it has had) the more likely this is to be the case. Open frames lack torsional rigidity and are particularly bad for this, as they are for most things.
(b) that the headset is correctly adjusted and in good condition. If it is too tight, the steering will tend to stick in one direction. The bars should turn freely without any resistance (other than friction between tyre and ground) but without any appreciable play. If the bearing surfaces are pitted, the bars will tend to turn with sort of click stops; that's bad too. If the bars are stiff on your new bike, adjust the headset and get a new mechanic. I can only surmise that ulumuri's new bike had the headset overtightened.
(c) that the wheels are true (and the tyres fitted straight).

The geometry of the frame will also be a significant factor. A frame with short trail - such as a typical track or road racing machine or an MTB - will be twitchier and more likely to develop a mind of its own - than a shallower angled tourer or hybrid.

As noted, life is easier if you are moving at a reasonable pace, and on a smooth surface. You should be in a low enough gear to be able to maintain your pace without pushing hard on the pedals, since the asymmetric forces will force you into a wobble.

Possibly the most important point is a corollary to ulumuri's first - almost all these factors apply to riding a bike normally. When you ride a bike, you should not steer with your hands; you steer by shifting your weight - in other words with your bottom. The bars are just there as somewhere to put the brake levers, to enable you to spread your weight out a bit and to prevent sudden fore-and-aft weight shifts when you go over bumps. If you ride like that, then taking your hands a few centimetres off the bars momentarily - without changing your body position - should be a doddle. The rest is just a matter of practising more of the same, for longer and further from the bars.

Don't stop pedaling

Or rather, the easiest way to pedal no-hands is to not alter your cadence at all. If you've worked up a nice rhythm, you're also probably seated in a particular position - the worst thing you can do is take your weight off the bars then alter the distribution of your weight on the saddle.

Also, if you have a road bike, try first letting go with one hand while riding in the drops - the lower position means a lower center of gravity, and gives you a few more seconds to grab the bars if you start to lose it.

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