The 3-week "rule" works well for most situations. You need to remember that bottle carbonation/conditioning is a secondary fermentation under conditions that are extremely stressful to the yeast. So they need to be given time to ferment, consume by-products and flocculate under anaerobic (i.e. they can't increase their numbers), high pressure and high-ethanol (toxic) conditions.
That said, if you are beyond the newbie stage characterized by an excessive fear of batches being ruined or infected based on tasting green beer, you don't have to follow the 3-week "rule" at all. First, keep the beer warm - mid- to upper 70s works great! Also, use your eyes - you can use one clear bottle per batch to use as an indicator to help you see the following processes more clearly: During bottle fermentation, the beer will turn somewhat opaque, scattering light shined through it. During flocculation, you'll notice a haze towards the bottom of the bottle. When the process is complete, the beer will become very transparent (assuming you don't have an inherently cloudy beer, or one that is very dark).
That's when you can take a couple of bottles, put them in the fridge for a couple of days and give them a taste. They'll probably be fine at this point. But as others have said, most beers will continue to improve for several weeks as they age, so let the rest sit warm until you are ready to drink them.