Users in consumption domains, like music, are often able to more efficiently provide preferences over a set of items (e.g. a playlist or radio) than over single items (e.g. songs). Unfortunately, this is an underexplored area of research, with most existing recommendation systems limited to understanding preferences over single items. Curating an item set exponentiates the search space that recommender systems must consider (all subsets of items!): this motivates conversational approaches-where users explicitly state or refine their preferences and systems elicit preferences in natural language-as an efficient way to understand user needs. We call this task conversational item set curation and present a novel data collection methodology that efficiently collects realistic preferences about item sets in a conversational setting by observing both item-level and set-level feedback. We apply this methodology to music recommendation to build the Conversational Playlist Curation Dataset (CPCD), where we show that it leads raters to express preferences that would not be otherwise expressed. Finally, we propose a wide range of conversational retrieval models as baselines for this task and evaluate them on the dataset.