How to shoot raw images with a Casio Exilim camera?

First, this depends on the exact model of Casio Exilim camera you have, because RAW is actually a collection of dozens of formats. And, they even differ between cameras of one series liek Casio Exilim.

Notes on the specifics of RAW support on some Casio Exilim models:

• If you have a Exilim PRO EX-F1 or Exilim EX-FH25, you're lucky, as these cameras support the open DNG format [source].
• The software dcraw (open source and available under Linux) can read and convert the format of the following Casio Exilim cameras [source]:
  • EX-S20
  • EX-S100
  • EX-Z4
  • EX-Z50
  • EX-Z55
  • EX-Z60
  • EX-Z75
  • EX-Z750
  • EX-Z850
  • EX-Z1050
  • EX-Z1080
  • Pro 505
  • Pro 600
  • Pro 700
• dcraw and ufraw failed to recognize and convert the RAW format of a Casio Exilim EX-S770 [source: personal experiment]. Up to now, I don't know of a way to read that format.
• No solution has been found so far to decode the RAW format images of a Casio Exilim EX-Z55
• The article fotofoto: Casio raw format files contains information on how to create and decode RAW files with the following cameras:
  • QV-2000UX
  • QV-3000EX
  • QV-3500EX
  • QV-4000
  • QV-5700
  • Exilim P600
  • Exilim P700
  • Exilim EX-Z55