Relation-based Argument Mining (RbAM) is the process of automatically determining agreement (support) and disagreement (attack) relations amongst textual arguments (in the binary prediction setting), or neither relation (in the ternary prediction setting). As the number of platforms supporting online debate increases, the need for RbAM becomes ever more urgent, especially in support of downstream tasks. RbAM is a challenging classification task, with existing state-of-the-art methods, based on Language Models (LMs), failing to perform satisfactorily across different datasets. In this paper, we show that general-purpose Large LMs (LLMs), appropriately primed and prompted, can significantly outperform the best performing (RoBERTa-based) baseline. Specifically, we experiment with two open-source LLMs (Llama-2 and Mistral) and with GPT-3.5-turbo on several datasets for (binary and ternary) RbAM, as well as with GPT-4o-mini on samples (to limit costs) from the datasets.