Apparently the search term I was missing was "Brownian motion".  With that, I found several leads.  They contradict each other somewhat, but I can at least post a partial answer:

Geisler - Sound to Synapse: Physiology of the Mammalian Ear:

> Estimates for the first of these sources, the pressure fluctuations due to the Brownian motion of air molecules impinging on the eardrum, are about 2 µPa (**−20 dB SPL**), when the frequency bandwidth relevant for the detection of a 3 kHz tone is included (Harris. 1968). Calculations using this number suggest that the behavioral thresholds of humans for 3 kHz tones are not limited by this Brownian motion, but that those for the most sensitive of cats may approach it (Green. 1976)

Dallos - The Auditory Periphery Biophysics and Physiology:

> By assuming a 1000-Hz bandwidth, Harris computed that the Brownian motion of air molecules generates a mean pressure fluctuation of 1.27×10<sup>−5</sup> dyne/cm<sup>2</sup> [**[−24 dB SPL][1]**]. The usually accepted value of sound pressure corresponding to free-field listening threshold is 18 dB above the pressure level of thermal  fluctuations. Thus one can immediately see that Brownian motion of air molecules is certainly not the limiting factor of our hearing sensitivity.

These both cite this paper that I don't have access to:

* Harris, G. G. Brownian motion in the cochlear partition. J Acoust. Soc. Am. 44: 176-186, 1968

But there's another available:

[Harris - Brownian motion and the threshold of hearing][2]:

> We can avoid the calculation of the Brownian noise at the eardrum by using the Brownian noise in a free field and comparing that with the minimum audible field (MAF) instead of the minimal audible pressure (MAP).
> 
> If we use frequency limits of 2500 Hz and 3500 Hz. we obtain a root mean square (rms) pressure fluctuation of 98 db below 1 dyne/cm<sup>2</sup> [**[−24 dB SPL][3]**]. The MAF<sup>2</sup> is about 80 db below 1 dyne/cm<sup>2</sup> at 3000 Hz. This is 18 db above the estimate of Brownian noise. It seems clear from this calculation that Brownian noise in the air is not a limiting factor to the threshold of hearing.

2.5 kHz to 3.5 kHz is not the total bandwidth that would be picked up by a microphone, though.

Yost & Killian - Hearing Thresholds: 

> By making some assumptions about the acoustic energy present in the Brownian motion of air molecules, it can be shown that a sound presented at 0 dB SPL is only 20-30 dB more intense than that being produced by Brownian motion

So **−20 to −30 dB SPL**.

Howard & Angus - Acoustics and Psychoacoustics:

> At 4kHz, which is about the frequency of the sensitivity peak, the pressure amplitude variations caused by the Brownian motion of air molecules, at room temperature and over a critical bandwidth, correspond to a sound pressure level of about **−23 dB**. Thus the human hearing system is close to the theoretical physical limits of sensitivity. In other words there would be little point in being much more sensitive to sound, as all we would hear would be a ”hiss” due to the thermal agitation of the air!

  [1]: http://www.wolframalpha.com/input/?i=20*log10%281.27%C3%9710%E2%81%BB%E2%81%B5%20dyne/cm%C2%B2%20/%20%2820%20%C2%B5Pa%29%29
  [2]: http://www.isa-audiology.org/periodicals/1962-1970_International_Audiology/InternatAudio,%20%20Vol.%207,%20%201968/No.%201%20%285-169%29/Harris,%20%20InternatAudio,%20%201968.pdf
  [3]: http://www.wolframalpha.com/input/?i=20*log10%281%20dyne/cm2/%2820%20%C2%B5Pa%29%29%20-%2098