Misner, Thorne, & Wheeler:
Oh yeah, and Jack Wisdom too. Science 299 5614 pp.1865-69 (10.1126/science.1081406) has an article by Wisdom on how a cyclically pulsating body could swim against space-time. Wisdom's explicit calculations involve a sort of mathematical jellyfish with three legs that change lengths and squeeze inwards in successive pulses. A relativistic inchworm.
Apparently it is possible to actually have a net translation in space simply by making cyclic changes in the shape of the object. The main problem with this is that the distance you would actually move is proportional to several things that are either really small (the size of the object and its relative movements), or are inversely proportional to things that are really big (the speed of light squared, and the distance from the body curving space-time). Wisdom calculates that a body of about a meter size, doing meter-sized cycles on the surface of the Earth would move about 10^-23 meters per cycle.
My back of the envelope thinking: assuming such an object could withstand a mechanical frequency of 200 Hz (not out of the range for high performance internal combustion engines), it would take just under 16 billion years to move its own length.
For this particular set of assumptions the amount of displacement is proportional to the mass creating the curvature and inversely proportional to the cube of one's distance from it. The trick, then, is to be really close to a black hole and start doing the shimmy. What about an extreme case, the super massive black hole at the center of our galaxy? This raises the central mass by a factor of about three trillion compared to Wisdom's Earth case. The distance change is trickier, because I fear that getting close will change some of the assumptions Wisdom made, since relativistic effects become important inside about 30 billion km from a black hole this big. Blindly I plough ahead (ignoring tidal effects on the apparatus), assume Wisdom's assumptions hold inside this distance (house of cards?) and can figure out that to equal or be greater than the translation effect on the Earth's surface, one needs to be closer than 100,000 km to the black hole. Unfortunately, this is also inside the Schwarzschild radius, or event horizon. No joy there.
Not a promising mode of locomotion in any neighbourhood, apparently.