This is apparently a result of flickr’s new People In Photos feature, which allows other flickr users to put names to faces in your photos. (If you don’t want people throwing your full name around on your flickr shots, you can disallow this for your photos here – but not for other people’s photos.)
There are a couple of workarounds – you can set your main gallery to Medium size photos and right-click from there; or you can click through to All Sizes and grab image URLs/HTML from there.
But there’s a quicker workaround, if you prefer: GustavoG has whipped up a Greasemonkey script that adds a small “URL” link under each photo. Voila! Show Image URL.
To use this, you must have Firefox with Greasemonkey installed.