In a structured prediction problem, we need to learn a predictor that can produce a structured output given a structured input (e.g., part-of-speech tagging). The key learning and inference challenge is due to the exponential size of the structured output space. This paper makes four contributions towards the goal of a computationally-efficient inference and training approach for structured prediction that allows to employ complex models and to optimize for non-decomposable loss functions. First, we define a simple class of randomized greedy search (RGS) based inference procedures that leverage classification algorithms for simple outputs. Second, we develop a RGS specific learning approach for amortized inference that can quickly produce high-quality outputs for a given set of structured inputs. Third, we plug our amortized RGS inference solver inside the inner loop of parameter-learning algorithms (e.g., structured SVM) to improve the speed of training. Fourth, we perform extensive experiments on diverse structured prediction tasks. Results show that our proposed approach is competitive or better than many state-of-the-art approaches in spite of its simplicity.