Results are reported from a search for the rare decays B_{s}^{0}→τ^{±}μ^{∓} and B^{0}→τ^{±}μ^{∓}, where the τ lepton is reconstructed in the channel τ^{-}→π^{-}π^{+}π^{-}ν_{τ}. These processes are effectively forbidden in the standard model, but they can potentially occur at detectable rates in models of new physics that can induce lepton-flavor-violating decays. The search is based on a data sample corresponding to 3 fb^{-1} of proton-proton collisions recorded by the LHCb experiment in 2011 and 2012. The event yields observed in the signal regions for both processes are consistent with the expected standard model backgrounds. Because of the limited mass resolution arising from the undetected τ neutrino, the B_{s}^{0} and B^{0} signal regions are highly overlapping. Assuming no contribution from B^{0}→τ^{±}μ^{∓}, the upper limit B(B_{s}^{0}→τ^{±}μ^{∓})<4.2×10^{-5} is obtained at 95% confidence level. If no contribution from B_{s}^{0}→τ^{±}μ^{∓} is assumed, a limit of B(B^{0}→τ^{±}μ^{∓})<1.4×10^{-5} is obtained at 95% confidence level. These results represent the first limit on B(B_{s}^{0}→τ^{±}μ^{∓}) and the most stringent limit on B(B^{0}→τ^{±}μ^{∓}).