The Cox proportional hazards model has become the model of choice to use in analyzing the effects of covariates on survival data. However, this assumption has significant restrictions on the behavior of the conditional survival function. The accelerated failure time model, which models the survival time and covariates directly through regression, provides an alternative approach to interpret the relationship between survival times and covariates. We consider here the estimation of the nonparametric regression function in the accelerated failure time model under right random censorship and investigate the asymptotic rates of convergence of estimators based on thresholding of empirical wavelet coefficients. We show that the estimators achieve nearly optimal minimax convergence rates within logarithmic terms over a large range of Besov function classes