Missing values in predictors are a common problem in survival analysis. In this paper, we review estimation methods for accelerated failure time models with missing predictors, and apply a new method called subsample ignorable likelihood (IL) Little and Zhang (J R Stat Soc 60:591-605, 2011) to this class of models. The approach applies a likelihood-based method to a subsample of observations that are complete on a subset of the covariates, chosen based on assumptions about the missing data mechanism. We give conditions on the missing data mechanism under which the subsample IL method is consistent, while both complete-case analysis and ignorable maximum likelihood are inconsistent. We illustrate the properties of the proposed method by simulation and apply the method to a real dataset.