The ratio of branching fractions R(D^{*-})≡B(B^{0}→D^{*-}τ^{+}ν_{τ})/B(B^{0}→D^{*-}μ^{+}ν_{μ}) is measured using a data sample of proton-proton collisions collected with the LHCb detector at center-of-mass energies of 7 and 8 TeV, corresponding to an integrated luminosity of 3  fb^{-1}. For the first time, R(D^{*-}) is determined using the τ-lepton decays with three charged pions in the final state. The B^{0}→D^{*-}τ^{+}ν_{τ} yield is normalized to that of the B^{0}→D^{*-}π^{+}π^{-}π^{+} mode, providing a measurement of B(B^{0}→D^{*-}τ^{+}ν_{τ})/B(B^{0}→D^{*-}π^{+}π^{-}π^{+})=1.97±0.13±0.18, where the first uncertainty is statistical and the second systematic. The value of B(B^{0}→D^{*-}τ^{+}ν_{τ})=(1.42±0.094±0.129±0.054)% is obtained, where the third uncertainty is due to the limited knowledge of the branching fraction of the normalization mode. Using the well-measured branching fraction of the B^{0}→D^{*-}μ^{+}ν_{μ} decay, a value of R(D^{*-})=0.291±0.019±0.026±0.013 is established, where the third uncertainty is due to the limited knowledge of the branching fractions of the normalization and B^{0}→D^{*-}μ^{+}ν_{μ} modes. This measurement is in agreement with the standard model prediction and with previous results.