We show theoretically that ultracold hydrogen atoms have very favorable properties for sympathetic cooling of molecules to microkelvin temperatures. We calculate the potential energy surfaces for spin-polarized interactions of H atoms with the prototype molecules NH(3Σ-) and OH(2Π) and show that they are shallow (50 to 80 cm(-1)) and only weakly anisotropic. We carry out quantum collision calculations on H+NH and H+OH and show that the ratio of elastic to inelastic cross sections is high enough to allow sympathetic cooling from temperatures well over 1 K for NH and around 250 mK for OH.