1. Open the Cover
Set the Kindle on a table or other secure surface and open up the cover. Note the metal nub sticking out about two thirds of the way from the top of the Kindle. This is the locking mechanism and the primary source of your frustration. You can't simply pop it out without damaging the cover and/or your e-reader.
2. Unlock the Latch
This is a two-part step. First, push that metal nub (or latch) down, in the direction of the bottom of the Kindle. It should slide down by a fraction of an inch. This releases the latching mechanism. Next, push the top of the Kindle horizontally, away from the metal nub.
3. The Bottom
At this point, the top of your e-reader is free, but the bottom slot of the Kindle is still attached to this devious crescent shaped hook.
The final step is to free the Kindle from that crescent hook. Simply rotate the e-reader down as shown and you're done. Your Kindle 3 is free of the cover.