Inter-annual crop yield variation is driven in large parts by climate variability, wherein the climate components of temperature and precipitation often play the biggest role. Nonlinear effects of temperature on yield as well as interactions among the climate variables have to be considered. Links between climate and crop yield variability have been previously studied, both globally and at regional scales, but typically with additive models with no interactions, or when interactions were included, with implications not fully explained. In this study yearly country level yields of maize, rice, soybeans, and wheat of the top producing countries were combined with growing season temperature and SPEI (standardized precipitation evapotranspiration index) to determine interaction and intensification effects of climate variability on crop yield variability during 1961-2014. For maize, soybeans, and wheat, heat and dryness significantly reduced yields globally, while global effects for rice were not significant. But because of interactions, heat was more damaging in dry than in normal conditions for maize and wheat, and temperature effects were not significant in wet conditions for maize, soybeans, and wheat. Country yield responses to climate variability naturally differed between the top producing countries, but an accurate description of interaction effects at the country scale required sub-national data (shown only for the USA). Climate intensification, that is consecutive dry or warm years, reduced yields additionally in some cases, however, this might be linked to spillover effects of multiple growing seasons. Consequently, the effect of temperature on yields might be underestimated in dry conditions: While there were no significant global effects of temperature for maize and soybeans yields for average SPEI, the combined effects of high temperatures and drought significantly decreased yields of maize, soybeans, and wheat by 11.6, 12.4, and 9.2%, respectively.