A little known piece of functionality on LinkedIn is the ability to export all of your connections to a CSV file. Amazingly this includes their email addresses as well as job title and a bunch of other info from their profiles.
You just need to go to your Connections page (My Network > Connections) and then click on the gear icon on the top right which will lead you a page called "Manage Sources". On the right hand column of this page you can see a "Export LinkedIn Connections" button.
Simple as that. Thanks LinkedIn!