Is the length of the rope known and constant? Is the anchor point of the rope fixed? If everything works out in your favour, you could probably get a reasonable estimate of the speed by measuring the average acceleration over a few seconds. If things aren't ideal - the rope length isn't fixed and know, or the center isn't stationary, the next approach I can see is to measure the acceleration in three dimensions and calculate the net acceleration vector, and use a gyro to measure the angular velocity. The maths will take some working out, but in theory it would be possible to determine the angular velocity excluding rotation around the axis of the rope, and given the angular velocity and acceleration it is possible to calculate the radius and speed.

It's going to involve some pretty heavy maths though, and unless you're familiar with calculations involving relative motion in multiple coordinate systems I think you'll find that rather challenging.