Thermally activated escape over a high barrier fluctuating randomly in time is investigated for asymmetric flipping rates. By means of the Fokker-Planck approach approximate methods developed for symmetric noise are applied to calculate the relaxation eigenvector and ultimate relaxation rate for piecewise linear potentials in the Smoluchowski limit. The ranges of validity of these approaches are discussed. Furthermore, the exact result for weak thermal noise but arbitrary barrier flipping rates is derived and its dependence on the asymmetry studied in detail.