It turns out that making the picture as a negative yields a much better result!
These are called lithophanes, and used to be made from porcelain, but work equally well in plastic!
I'm using the program Photo to Mesh to lift a 3D picture from a 2D photo. There are a few programs out there that will do the same, but this one was cheap and easy to use. A free demo is available too.
Here are a couple samples I have done. Printed in white plastic .
These are simply held up the light to show the detail. The tops get a little rough because they are printed vertically, and the print tends to wobble around a bit as it gets higher. The thinnest parts are .6mm so its not a surprise.
Here is what the puppy looks like without backlight.