Despite the predominance of English in their training data, English-centric Large Language Models (LLMs) like GPT-3 and LLaMA display a remarkable ability to perform multilingual tasks, raising questions about the depth and nature of their cross-lingual capabilities. This paper introduces the decomposed prompting approach to probe the linguistic structure understanding of these LLMs in sequence labeling tasks. Diverging from the single text-to-text prompt, our method generates for each token of the input sentence an individual prompt which asks for its linguistic label. We assess our method on the Universal Dependencies part-of-speech tagging dataset for 38 languages, utilizing both English-centric and multilingual LLMs. Our findings show that decomposed prompting surpasses the iterative prompting baseline in efficacy and efficiency under zero- and few-shot settings. Further analysis reveals the influence of evaluation methods and the use of instructions in prompts. Our multilingual investigation shows that English-centric language models perform better on average than multilingual models. Our study offers insights into the multilingual transferability of English-centric LLMs, contributing to the understanding of their multilingual linguistic knowledge.