The proportional hazards (PH) model is the most widely used semiparametric regression model for analyzing right-censored survival data based on the partial likelihood method. However, the partial likelihood does not exist for interval-censored data due to the complexity of the data structure. In this paper, we focus on general interval-censored data, which is a mixture of left-, right-, and interval-censored observations. We propose an efficient and easy-to-implement Bayesian estimation approach for analyzing such data under the PH model. The proposed approach adopts monotone splines to model the baseline cumulative hazard function and allows to estimate the regression parameters and the baseline survival function simultaneously. A novel two-stage data augmentation with Poisson latent variables is developed for the efficient computation. The developed Gibbs sampler is easy to execute as it does not require imputing any unobserved failure times or contain any complicated Metropolis-Hastings steps. Our approach is evaluated through extensive simulation studies and illustrated with two real-life data sets.