We present a measurement of the partial branching fractions and mass spectra of the exclusive radiative penguin processes B{yields}K{pi}{pi}{gamma} in the range m{sub K{pi}}{sub {pi}}<1.8 GeV/c{sup 2}. We reconstruct four final states: K{sup +}{pi}{sup -}{pi}{sup +}{gamma}, K{sup +}{pi}{sup -}{pi}{sup 0}{gamma}, K{sub S}{sup 0}{pi}{sup -}{pi}{sup +}{gamma}, and K{sub S}{sup 0}{pi}{sup +}{pi}{sup 0}{gamma}, where K{sub S}{sup 0}{yields}{pi}{sup +}{pi}{sup -}. Using 232x10{sup 6} e{sup +}e{sup -}{yields}BB events recorded by the BABAR experiment at the SLAC PEP-II asymmetric-energy storage ring, we measure the branching fractions B(B{sup +}{yields}K{sup +}{pi}{sup -}{pi}{sup +}{gamma})=[2.95{+-}0.13(stat){+-}0.20(syst)]x10{sup -5}, B(B{sup 0}{yields}K{sup +}{pi}{sup -}{pi}{sup 0}{gamma})=[4.07{+-}0.22(stat){+-}0.31(syst)]x10{sup -5}, B(B{sup 0}{yields}K{sup 0}{pi}{sup +}{pi}{sup -}{gamma})=[1.85{+-}0.21(stat){+-}0.12(syst)]x10{sup -5}, and B(B{sup +}{yields}K{sup 0}{pi}{sup +}{pi}{sup 0}{gamma})=[4.56{+-}0.42(stat){+-}0.31(syst)]x10{sup -5}.
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