Sensation of motion can come through three of the physical senses. Each of the three senses is successively unavailable as we move through the examples of an amusement park ride, the plummeting elevator, and AF447.
This sense is fairly well understood, so I'll limit this bullet point to pointing out that the visual sense is correlated in the human brain to the vestibular sense. In the amusement park ride, the inputs recieved from the vestibular sense and vision match, and there is no resulting confusion about the perceived motion. In cases where those two senses are decoupled, it is common for nausea to result. In the elevator and airplane in instrument conditions, a visual sense grounding you to a world reference frame is denied. This is one of the senses which can detect unaccelerated motion, more on "unaccelerated" below.
Here we have information which comes in through our sense of touch, primarily through nerve endings in the skin. The roller coaster comes through here again by exposing you to the elements giving you a "rush of wind" in your face as you move through what may be a stationary mass of air. In the closed environments of the elevator and cockpit, this is again denied.
Here is where things get interesting. For the purposes of this discussion, I'll add to the vestibular senses the "seat of the pants", "pit of the stomach" and other bodily means of detecting acceleration. Several other answers have pointed out that all of these senses are sensitive not to velocity but to acceleration, which high school physics reminds us is the the time rate of change, or derivative, of velocity. This means that if velocity is not changing we are not going to have a sensation of motion, although in a gravity field, we may have some sense of orientation (you can have a sense of hanging upside down or laying sideways). There is no difference to these senses between sitting on the ground with the engines running, cruising at 500 knots in level flight, or being in a 500,1000, or 3000 foot per minute descent. As long as velocity is not changing, these senses cannot tell the difference. This is the problem with AF447, No visual, no touch, and in unaccelerated descent, no vestibular sense.
If the airdata instruments are offline, there is nothing to indicate the descent until the cabin pressure starts to increase, putting some pressure on your eardrums. That also won't happen until the aircraft descends to a point where the outside air pressure increases to a point above the cabin pressure then continues to build to a point that you notice. They were doomed.
There is another problem, even with detecting acceleration. The problem is connected to the human brain's remarkable ability to filter out stimuli. A normal human brain is capable of adjusting to the tint of your sunglasses, ignoring conversations in a crowded room, and is very good at not notifying the conscious thought center of the pressure your elastic socks are putting on your ankles. Indeed, it can even lose track of constant accelerations. If you live near a playa, you can do a blind test by driving your car out and enter a gentle turn giving a tenth of a "g" of lateral acceleration. Have someone else do the driving. I am here to tell you that the resulting sensation can be filtered out. Even our sense of acceleration can be lost under constant acceleration. In physics, there is yet another level of derivative, the time rate of change of acceleration, called "jerk", which is harder to not detect.
For more information on the brain's filtering capability, look up the "Reticular Activating System". Understanding it can help you be more safe, and also help you reach your goals.