A flexure bearing unit in a linear drive compressor that supports a piston on a yoke for reciprocating movement within a housing includes a plurality of flat discs each provided with a central opening through which extends the piston and a plurality of outer openings receiving parts of the yoke. The flexure bearing is provided with a plurality of first supports each disposed between adjacent flat discs at the outer peripheral portions of the flat discs for supporting opposite sides of the flat discs at the outer peripheral portions of the flat discs. The flexure bearing is also provided with a plurality of second supports each provided between adjacent flat discs at the inner peripheral portions of the flat discs for supporting opposite sides of the flat discs at the inner peripheral portions of the flat discs. The flat discs each include a plurality of spirally extending slits which each possess oppositely located ends. One end of each slit terminates in the outer peripheral portion of the flat disc that is supported on opposite sides by the first supports and the other end of each slit terminates in the inner peripheral portion of the flat disc that is supported on opposite sides by the second supports.