One of the main science objectives for the Laser Interferometer Space Antenna (LISA) is to quantitatively map the strong field regions around compact objects using Extreme-Mass-Ratio Inspirals (EMRIs). This idea has been shown to be possible in principle, however in practice only inspirals in a Kerr spacetime have been studied in detail. A spacetime mapping algorithm for an EMRI inspiral into a generic compact object is formulated using ideas from integrable systems. I discuss several aspects of the theoretical development required to make the problem tractable. Some recent results about particle orbits around "Bumpy Black" holes are highlighted.