Okay so I know about half of how to do this. So I made a plane, then converted it to a path with Alt-C. Then I made another plane and made many loop cuts with Ctrl-R and the scroll wheel. Then I used the curve modifier to make the non-curve plane follow the curve plane. But it cuts corners:
It could be that you have to set the curve handles to to the Vector Type using the v key to bring up the menu. You can change the handle type individually or all at once depending on what is selected.
I am just guessing here since you did not upload a .blend file to PasteAll.org with the link pasted here.