Numerous text classification tasks inherently possess hierarchical structures among classes, often overlooked in traditional classification paradigms. This study introduces novel approaches for hierarchical text classification using Large Language Models (LLMs), exploiting taxonomies to improve accuracy and traceability in a zero-shot setting. We propose two hierarchical classification methods, namely (i) single-path and (ii) path-traversal, which all leverage the hierarchical class structures inherent in the target classes (e.g., a bird is a type of animal that belongs to a species) and improve naïve hierarchical text classification from literature. We implement them as prompts for generative models such as OpenAI GPTs and benchmark them against discriminative language models (BERT and RoBERTa). We measure the classification performance (precision, recall, and F1-score) vs. computational efficiency (time and cost). Throughout the evaluations of the classification methods on two diverse datasets, namely ComFaSyn, containing mental health patients' diary entries, and DBpedia, containing structured information extracted from Wikipedia, we observed that our methods, without any form of fine-tuning and few-shot examples, achieve comparable results to flat classification and existing methods from literature with minimal increases in the prompts and processing time.