In this paper, we study differentially private empirical risk minimization (DP-ERM). It has been shown that the (worst-case) utility of DP-ERM reduces as the dimension increases. This is a major obstacle to privately learning large machine learning models. In high dimension, it is common for some model's parameters to carry more information than others. To exploit this, we propose a differentially private greedy coordinate descent (DP-GCD) algorithm. At each iteration, DP-GCD privately performs a coordinate-wise gradient step along the gradients' (approximately) greatest entry. We show theoretically that DP-GCD can improve utility by exploiting structural properties of the problem's solution (such as sparsity or quasi-sparsity), with very fast progress in early iterations. We then illustrate this numerically, both on synthetic and real datasets. Finally, we describe promising directions for future work.