We present ASPIRO, an approach for structured data verbalisation into short template sentences in zero to few-shot settings. Unlike previous methods, our approach prompts large language models (LLMs) to directly produce entity-agnostic templates, rather than relying on LLMs to faithfully copy the given example entities, or validating/crafting the templates manually. We incorporate LLM re-prompting, triggered by algorithmic parsing checks, as well as the PARENT metric induced consistency validation to identify and rectify template generation problems in real-time. ASPIRO, compared to direct LLM output, averages 66\% parsing error rate reduction in generated verbalisations of RDF triples on the DART dataset. Our best 5-shot text-davinci-003 setup, scoring BLEU of 50.62, METEOR of 45.16, BLEURT of 0.82, NUBIA of 0.87, and PARENT of 0.8962 on the Rel2Text dataset, competes effectively with recent fine-tuned pre-trained language models.