Prediction of drug combination responses is a research question of growing importance for cancer and other complex diseases. Current machine learning approaches generally consider predicting either drug combination synergy summaries or single combination dose-response values, which fail to appropriately model the continuous nature of the underlying dose-response combination surface and can lead to inconsistencies when a synergy score or a dose-response matrix is reconstructed from separate predictions. We propose a structured prediction method, comboKR, that directly predicts the drug combination response surface for a drug combination. The method is based on a powerful input-output kernel regression technique and functional modeling of the response surface. As an important part of our approach, we develop a novel normalisation between response surfaces that standardizes the heterogeneous experimental designs used to measure the dose-responses, and thus allows training the method with data measured in different laboratories. Our experiments on two predictive scenarios highlight the suitability of the proposed approach especially in the traditionally challenging setting of predicting combination responses for new drugs not available in the training data.