  Anomalous Microwave Emission (AME) has been previously studied in two
well-known molecular clouds and is thought to be due to electric dipole
radiation from small spinning dust grains. It is important to measure the
polarization properties of this radiation both for component separation in
future cosmic microwave background experiments and also to constrain dust
models. We have searched for linearly polarized radio emission associated with
the $\rho$ Ophiuchi and Perseus molecular clouds using {\it WMAP} 7-year data.
We found no significant polarization within an aperture of $2^{\circ}$
diameter. The upper limits on the fractional polarization of spinning dust in
the $\rho$ Ophiuchi cloud are 1.7%, 1.6% and 2.6% (at 95% confidence level) at
K-, Ka- and Q-bands, respectively. In the Perseus cloud we derived upper limits
of 1.4%, 1.9% and 4.7%, at K-, Ka- and Q-bands, respectively; these are similar
to those found by L\'opez-Caraballo et al. If AME at high Galactic latitudes
has a similarly low level of polarization, this will simplify component
separation for CMB polarization measurements. We can also rule out single
domain magnetic dipole radiation as the dominant emission mechanism for the
20-40 GHz. The polarization levels are consistent with spinning dust models.
