Somehow I must have missed this popular science initiative from Rob Bryanton. It's been around since at least 2006, backed by a book, videos, website, blog, video blog, interactive media chat-room, and even cheesy songs written and performed by Rob.

Here's an introductory video that walks through imagining ten dimensions:

This animation illustrates the concepts presented in chapter one of the book "Imagining the Tenth Dimension" by Rob Bryanton.