Reliable cost effective effort estimation remains a considerable challenge for software projects. Recent work has demonstrated that the popular Planning Poker practice can produce reliable estimates when undertaken within a software team of knowledgeable domain experts. However, the process depends on the availability of experts and can be time-consuming to perform, making it impractical for large scale or open source projects that may curate many thousands of outstanding tasks. This paper reports on a full study to investigate the feasibility of using crowd workers supplied with limited information about a task to provide comparably accurate estimates using Planning Poker. We describe the design of a Crowd Planning Poker (CPP) process implemented on Amazon Mechanical Turk and the results of a substantial set of trials, involving more than 5000 crowd workers and 39 diverse software tasks. Our results show that a carefully organised and selected crowd of workers can produce effort estimates that are of similar accuracy to those of a single expert.