The basic theory and design of a reciprocating magnetic refrigerator to pump heat from 2.2 to 4.2 K is presented. The results of initial experiments are shown. These results include conduction losses, eddy current losses, frictional losses, and mixing losses. Two cooling cycles were attempted and a net cooling power of 52 mW was observed at 1/60 Hz. The key problems in this design are identified and discussed.