  Millimeter wavelength polarimetry of accreting black hole systems can provide
a tomographic probe of the accretion flow on a wide range of linear scales. We
searched for linear polarization in two low luminosity active galactic nuclei
(LLAGN), M81 and M84, using the Combined Array for Millimeter Astronomy (CARMA)
and the Submillimeter Array (SMA). We find upper limits of $\sim 1 - 2\%$
averaging over the full bandwidth and with a rotation measure (RM) synthesis
technique. These low polarization fractions, along with similar low values for
LLAGN M87 and 3C84, suggest that LLAGN have qualitatively different
polarization properties than radio-loud sources and Sgr A*. If the sources are
intrinsically polarized and then depolarized by Faraday rotation then we place
lower limits on the RM of a few times $10^7\, {\rm rad\, m^{-2}}$ for the full
bandwidth case and $\sim 10^9\, {\rm rad\, m^{-2}}$ for the RM synthesis
analysis. These limits are inconsistent with or marginally consistent with
expected accretion flow properties. Alternatively, the sources may be
depolarized by cold electrons within a few Schwarzschild radii from the black
hole, as suggested by numerical models.
