Hmm, I don't know. That all sounds right. The airlock going off after you transferred it into the secondary was just CO2 coming out of solution. Leaving it sit for three weeks in a primary or secondary shouldn't be a problem at all.
I would think that after three weeks in the bottle you would have at least some carbonation.
Did you stir the beer well when you added the priming sugar into the bottling bucket? Have you randomly sampled a number of the bottles? Is the problem consistent across the batch?
You know you have to cap the bottles right?
....Just kidding, but are you sure that the caps are nice and tight? Submerge one in a bucket of water and look for bubbles to see if gas is escaping.