Typically, the intensity noise of solid-state lasers is dominated by a resonant relaxation oscillation, RRO, at intermediate frequencies (kilohertz to megahertz) and by pump-source noise at frequencies below the RRO. The RRO is driven by vacuum fluctuations as well as by pump-source fluctuations and is therefore present irrespective of the pump-source noise level. However, the intensity noise at frequencies below the RRO can be substantially lowered by use of a low-noise pump source. This behavior is experimentally studied for diode-pumped Nd:YAG ring lasers. An experimental comparison is made between pumping with a single-element diode laser (SEDL) or with a diode-laser array (DLA). We find good agreement with theory for the SEDL but not for the DLA because the DLA's output intensity noise is spatially variant. We also show that pump-source frequency noise has only a minor effect on the intensity noise of the Nd:YAG laser. The requirements for low-noise operation of solid-state lasers are discussed.