A apparatus for low AC loss thermal shielding includes a plurality of thermally conducting fibers (70, 88, 158, 168, 184) positioned along a desired direction of heat conduction. The fibers (70, 88, 158, 168, 184) are electrically insulated from each other. The fibers (70, 88, 158, 168, 184) are bonded together with a matrix (110, 170), and a thermal link (126) connects the bonded fibers (70, 88, 158, 168, 184) to a cryogenic cold head (144).