Missing covariate data often arise in biomedical studies, and analysis of such data that ignores subjects with incomplete information may lead to inefficient and possibly biased estimates. A great deal of attention has been paid to handling a single missing covariate or a monotone pattern of missing data when the missingness mechanism is missing at random. In this article, we propose a semiparametric method for handling non-monotone patterns of missing data. The proposed method relies on the assumption that the missingness mechanism of a variable does not depend on the missing variable itself but may depend on the other missing variables. This mechanism is somewhat less general than the completely non-ignorable mechanism but is sometimes more flexible than the missing at random mechanism where the missingness mechansim is allowed to depend only on the completely observed variables. The proposed approach is robust to misspecification of the distribution of the missing covariates, and the proposed mechanism helps to nullify (or reduce) the problems due to non-identifiability that result from the non-ignorable missingness mechanism. The asymptotic properties of the proposed estimator are derived. Finite sample performance is assessed through simulation studies. Finally, for the purpose of illustration we analyze an endometrial cancer dataset and a hip fracture dataset.