In this article we describe the background challenges for the CUORE experiment posed by surface contamination of inert detector materials such as copper, and present three techniques explored to mitigate these backgrounds. Using data from a dedicated test apparatus constructed to validate and compare these techniques we demonstrate that copper surface contamination levels better than 10E-07 - 10E-08 Bq/cm2 are achieved for 238U and 232Th. If these levels are reproduced in the final CUORE apparatus the projected 90% C.L. upper limit on the number of background counts in the region of interest is 0.02-0.03 counts/keV/kg/y depending on the adopted mitigation technique.