For many applications, reducing sample resistance, rather than increasing probe Q or filling factor, is the only way to further improve the signal-to-noise ratio of cryogenically cooled NMR probes. In this paper, bounds are calculated for the minimum sample resistance that can be achieved for various sample geometries. The sample resistance of 100 mM NaCl in H(2)O in 5 mm sample tubes was measured on a 600 MHz cold probe to be within 14% of the optimum value. The minimum sample resistance can however be lowered by altering the tube cross section. Rectangular tubes oriented with the long axis along the RF magnetic field are particularly favourable.