How can I empty my Camera Roll from my iPhone? I don't want to save anything—just wipe out the Roll like one would reformat a digital camera's memory card. Is there any easy non-computer route? If there aren't any system settings, do any apps do this?

Assume I have too many to delete individually (1,000+).
I'm afraid there isn't an easy way to do so. As far as I know, there is no app which can do what you want... So I've come up with two alternative solutions.

The most easy one is to connect your iPhone to your Mac, launch Image Capture (which comes by default on your Mac), select all the photos and delete them. Here is a tutorial how to do so. 

You can do this on your iPhone as well, without a Mac, but you'll have to do it manually. Which isn't a grateful job if you've got over a 1,000 photos...   

Open the camera app, hit the upper-right icon (share), tap all the photos you want to delete (they are marked with a red check-icon) and then hit delete.