Regression quantiles can be underpowered or biased when there are miss- ing values in some covariates. We propose a method that produces consistent linear quantile estimation in the presence of missing covariates. The proposed method cor- rects bias by constructing unbiased estimating equations that simultaneously hold at all the quantile levels. It utilizes all the available data, and produces uniformly consistent estimators. An iterative EM-type algorithm is provided for solving the estimating equations. The finite sample performance of the method is investigated in a simulation study. Finally, the methodology is applied to data from the National Health and Nutrition Examination Survey.