Cross-lingual named entity recognition (NER) suffers from data scarcity in the target languages, especially under zero-shot settings. Existing translate-train or knowledge distillation methods attempt to bridge the language gap, but often introduce a high level of noise. To solve this problem, consistency training methods regularize the model to be robust towards perturbations on data or hidden states. However, such methods are likely to violate the consistency hypothesis, or mainly focus on coarse-grain consistency. We propose ConNER as a novel consistency training framework for cross-lingual NER, which comprises of: (1) translation-based consistency training on unlabeled target-language data, and (2) dropoutbased consistency training on labeled source-language data. ConNER effectively leverages unlabeled target-language data and alleviates overfitting on the source language to enhance the cross-lingual adaptability. Experimental results show our ConNER achieves consistent improvement over various baseline methods.