There are several ways to present the answer to this question. I'll start with one suggested by your wording.
Einstein does say that gravity and acceleration have the same effects. More precisely, he says that there are no differences between the effects of adding a uniform gravitational field or of switching to coordinates that uniformly accelerate with respect to your old coordinates. Those effects depend on the direction of the gravity or of the acceleration. For the right directions and magnitudes, the gravity effects and the acceleration effects exactly cancel. That's what happens in free fall.
Another way to look at it is to ask how you feel forces. You (or some instrument) feel forces because they push and pull differently on different parts of a nerve, or a strain gauge, or something, squeezing them into slightly different states. As Galileo discovered, in a uniform gravitational field all objects fall in exactly the same way, assuming no other forces are present. Every little part falls along with every other part. That means that in free fall there are no symptoms of how fast you're falling, or whether there's any gravity present.
When you think you feel gravity, what you are really feeling are other forces- say forces from the floor or a chair. These forces are almost always electromagnetic in origin. Notice that you feel them in particular parts of your body, even though gravity acts uniformly on all parts of your body.
Actually, there are tiny tidal forces on objects with nonzero extent. Gravitational forces vary in strength and direction as you move around in space, and the difference between the gravitational force in one place and in another can be registered on strain gauges (the Earth's oceans, and the Earth itself act as a gigantic tidal strain gauge as the moon orbits above and as we orbit the sun). On a smaller scale, two small objects (like marbles) set adrift in the space shuttle may have a very small difference in the gravitational forces on them. Each is separately in orbit around the earth. If one is a tiny bit closer to the earth's center, it should make its orbit in a tiny bit less time than one that's farther away, and to an observer floating with them on the space shuttle, he should see one of them slowly creeping across the cabin relative to the other one.
In practice, it's hard to get the initial momentum of a floating object so low that the tidal forces are visible for a small area like the interior of the space shuttle. But it is tidal forces like these which tear objects apart as they fall into black holes.
(published on 10/22/2007)