Event Causality Identification (ECI), which aims to detect whether a causality relation exists between two given textual events, is an important task for event causality understanding. However, the ECI task ignores crucial event structure and cause-effect causality component information, making it struggle for downstream applications. In this paper, we introduce a novel task, namely Event Causality Extraction (ECE), aiming to extract the cause-effect event causality pairs with their structured event information from plain texts. The ECE task is more challenging since each event can contain multiple event arguments, posing fine-grained correlations between events to decide the cause-effect event pair. Hence, we propose a method with a dual grid tagging scheme to capture the intra- and inter-event argument correlations for ECE. Further, we devise a event type-enhanced model architecture to realize the dual grid tagging scheme. Experiments demonstrate the effectiveness of our method, and extensive analyses point out several future directions for ECE.