Survival analysis for randomized censored data is common in clinical trial. Among various survival models proposed to deal with such censoring, Weilbull regression model is widely used as one of accelerated failure time models, because it's flexible to suit different applications. It's well known that Bayesian analysis has the advantage in dealing with censored data and small sample over frequentist methods. Therefore, this paper presents the Weibull regression model for randomized censored data from Bayesian perspective, and then computes the Bayesian estimator based on the Markov Chain Monte Carlo (MCMC) method. The Gibbs sampling is proposed to simulate the Markov chain of parameters' posterior distribution dynamically, which avoids the calculation of complex integrals of the posterior distribution effectively. Finally the simulation with real clinical data of lymph sarcoma is presented. The whole procedure is implemented by the freely available software WinBUGS.