We present the first results using the Reuven Ramaty High-Energy Solar Spectroscopic Imager (RHESSI) to observe solar X-ray emission not associated with active regions, sunspots, or flares (the quiet Sun). Using a newly developed chopping technique (fan-beam modulation) during seven periods of offpointing between 2005 June and 2006 October, we obtained upper limits over 3-200 keV for the quietest times when the GOES 12 1-8 Å flux fell below 10-8 W m-2. These values are smaller than previous limits in the 17-120 keV range and extend them to both lower and higher energies. The limit in 3-6 keV is consistent with a coronal temperature <=6 MK. For quiet-Sun periods when the GOES 12 1-8 Å background flux was between 10-8 and 10-7 W m-2, the RHESSI 3-6 keV flux correlates to this as a power law, with an index of 1.08+/-0.13. The power-law correlation for microflares has a steeper index of 1.29+/-0.06. We also discuss the possibility of observing quiet-Sun X-rays due to solar axions and use the RHESSI quiet-Sun limits to estimate the axion-to-photon coupling constant for two different axion emission scenarios.