As this was long overdue, I decided to make a head unit install DIY for the 03-04 Civic. This may apply to other years as well, but AFAIK, it is only for the 03-04. This DIY was created using an 04 LX coupe.
1) Stick your hand under the trim ring and pull
2) The ring comes off easily
3) Stick your finger in the location shown and pull up HARD. It can be very stiff but it does come out. There is a hole there to stick your finger in and make it easier.
4) The console piece does come off but it can be a pain in the ass sometimes. If it seems stuck in one part, look very closely to where the clip is (the console piece will be cut out a little just above it), stick a very small screwdriver and try to push the clip out. It will come out eventually so be persistent.
5) Use a phillips head screwdriver to remove the 2 screws located near the cup holders on either side of the console
6) Stick your hand under the "horseshoe shaped piece" as shown and pull it out. It is simply held in by clips, 2 on each side. Pull the sides out first and then pull the entire piece out by pulling up and back. Unplug the cigarette lighter and set this piece aside.
7) This is what your dash should now look like, and you can start taking the head unit out.
8) Remove these 2 screws to allow you to take the panel containing your HVAC controls, hazard light and head unit to come out. You can use either a very short phillips head screwdriver or an 8mm socket and ratchet. I highly recommend the ratchet. One screw is in the center and one is towards the right, but they are up underneath the dash and you may have to get under there at a wierd angle to see them, but they are very easy to get out using a ratchet.
9) Stick your hand behind the metal bar that you just took the screws out of and push it kind of up and forward. The top of this bar is part of the panel containing the head unit and the bottom half is part of the car. Push on the bar so that the head unit pops out. It may be stiff if it hasn't been removed before but it does come out.
That's all there is to it, and reassembly is the exact opposite of disassembly.
If you get stuck, of course feel free to post in the topic.