If you’re interested in the final, working solution, skip to the end of this post (the part where it says “UPDATE UPDATE UPDATE” :).
You’ve made your award-winning illustration in Visio 2003, and now you’d like to include it in your award-winning LaTeX proposal. You find this page with tips, but you’ve also discovered that none of them work for you, or that you don’t have the necessary software, or that they half-work in a very frustrating fashion (this was the case for me with visio -> emf -> openoffice draw -> eps and borked transparencies).