In many surveys and clinical trials, we obtain measurements on covariates or biomarkers that are left-censored due to the limit of detection. In such cases, it is necessary to correct for the left-censoring when studying covariate effects in regression models. The expectation-maximization (EM) algorithm is widely used for the likelihood inference in generalized linear models with censored covariates. The EM method, however, requires intensive computation involving high-dimensional integration with respect to the covariates when the dimension of the censored covariates is large. To reduce such computational difficulties, we propose and explore a Monte Carlo EM method based on the Metropolis algorithm. The finite-sample properties of the proposed estimators are studied using Monte Carlo simulations. An application is also provided using actual data obtained from a health and nutrition examination survey.