  Annihilation of dark matter can result in the production of stable Standard
Model particles including electrons and positrons that, in the presence of
magnetic fields, lose energy via synchrotron radiation, observable as radio
emission. Galaxy clusters are excellent targets to search for or to constrain
the rate of dark matter annihilation, as they are both massive and dark matter
dominated. In this study, we place limits on dark matter annihilation in a
sample of nearby clusters using upper limits on the diffuse radio emission, low
levels of observed diffuse emission, or detections of radio mini-haloes. We
find that the strongest limits on the annihilation cross section are better
than limits derived from the non-detection of clusters in the gamma-ray band by
a factor of approximately 3 or more when the same annihilation channel and
subtructure model, but different best-case clusters, are compared. The limits
on the cross section depend on the assumed amount of substructure, varying by
as much as 2 orders of magnitude for increasingly optimistic substructure
models as compared to a smooth NFW profile. In our most optimistic case, using
the results of the Phoenix Project (Gao et al. 2012b) we find that the derived
limits reach below the thermal relic cross section of 3x10^-26 cm^3 s^-1 for
dark matter masses as large as 400 GeV, for the bbar annihilation channel. We
discuss uncertainties due to the limited available data on the magnetic field
structure of individual clusters. We also report the discovery of diffuse radio
emission from the central 30-40 kpc regions of the groups M49 and NGC4636.
