My new car, which needed an AUX input to the stereo
Update 12/05/04: My instructions for radio removal had the keys facing the wrong way.
Update 12/27/04: Added link to VW stereo removal instructions and section for Pat's mods.
Update: 2/20/05: Added additional information from Lee Hassig on specific parts.
Update: 7/22/06: Fixed some links to enfig.
So, I bought a new car: a 2005 Volkswagen Golf. This replaces the 1994 Chrysler Concorde (which, ironically is still sitting in the driveway, because I'm waiting for my neighbor to buy it. Oh well, it's typical Rhode Island to have a boat in the driveway, right?)

Old car on the left, new car on the right. An improvement, eh?
The Problem:
This is the GLS model, with the Monsoon Sound package. I have a Rio Karma and the cassette deck adapter just doesn't cut it on this nice stereo. There's audible artifacts that cause me great annoyance and displeasure. I'd just do an stereo jack in, except that the stereo doesn't have an input for that.
Update - 7/22/06: The Rio Karma has died, so I will be using an iPod Nano with RockBox firmware for all my ogg playing needs. The hookup remains the same, which is the nice thing about this setup. Any player with a 1/8" audio jack works.
The Solution:
One thing the stereo does have is an input for a CD changer. I didn't get a changer, because I don't need one. However, the input is still there. One could, in theory, use the inputs, right? Well, the head unit expects to have a device attached to it that it can talk to. So, one needs something to emulate a CD changer, right? A BlitzSafe Aux Input does the trick quite nicely, and Christian at Enfig Car Stereo was quite helpful. A set of stereo removal keys (note, these are different than the ones I used, so the ones in the pics will be a little different. However, they should work just fine) and a 3 foot 1/8 inch stereo to RCA cable later, I was good to go. (I got the BlitzSafe that plugs in to the head unit, there's another one that connects to the harness for the CD changer in the trunk, if you wanted to do a PC or something back there). Also, note that I can't take credit for this, I got the idea from this person.
Pulling the old Head Unit:
This was pretty easy, once I figured out which way the keys go. Use the longer prongs, with the notched end facing the inside of the stereo. Put all 4 keys in; you'll hear them snap in to place after about half an inch. The stereo should then come out freely. If it doesn't, you're doing something wrong. These instructions may explain it a little more clearly. Anyway, a little bit of work later, and this is the result:
I sat in the driver's seat, pushed the steering wheel all the way in and the seat back a bit in order to make room. When I pulled out the unit, I twisted it to the left so that I could get at the back and disconnect the wires. Note that, if you're careful, you don't need to disconnect everything. In fact, if you don't disconnect the power, you don't have to punch in the unlocking code. So, all you need to do is to disconnect the antenna (so that you can twist the unit around) and the CD input (because you're going to plug something in where it used to be, duh). Also worth noting is that the CD wiring harness had a blue retention clip on it. I removed the clip before I took the picture (because I forgot), but it will be there. Once you've unplugged the CD changer wiring bundle, use a small screwdriver to pop off the blue clip - you'll need it.
Aha! Here's the blue clip. I've attached it to the BlitzSafe adapter, but you can't really tell. Pop it off the old harness, pop it on to the new one.
Jacking In
The BlitzSafe adapter has female RCA jacks, so they get connected to the 1/8 to RCA adapter, like so:
I don't like my connectors coming apart on me, so a little bit of electrical tape (you don't really need a link for that, do you?) later, we have this marvelous work of art:
Finding Useful Holes
Now, the astute reader might be asking "How are you going to get the cable out from behind the head unit"? Well, poking around the interior, I found that, on the bottom of the console for the '05 Golf (and presumably everything since '02) there is a trim piece that pops off. Underneath that piece, on the left, is a hole that goes back into the console. A little bit of fishing later, we have:

(Note the hole by the badly drawn arrow)
Finishing up
Stuffing all the wires back in, and fitting the BlitzSafe adapter in wherever it fits allows us to have enough space to put the head unit back in. (Lee adds a note here that it is very important to get any wires clear of the hole for the head unit. It's probably easiest to pull them through the output hole, then install the head unit, then stuff them back in).
One very important thing: there is a ground connection. If that isn't connected, then the device will not work. That ground is used as a logic ground, and without it the digital circuity won't function, meaning that the head unit won't think there's a CD changer and won't allow you to select that input source. A 5mm nut (thanks to Lee for the dimension) and a lock washer later, we have a solid ground to the grounding post on the back of the stereo:
All done
Slide the unit back in and it clicks in to place. The Karma (or any other appropriate input device) just plugs in down at the bottom.
And, without the Rio connected, we see that we don't see much of anything, just a little tail out the bottom.
Addendum
Since the Rio Karma's display is backlit in blue, it looks crispy in the dark.
Another approach
Thanks to Pat Masterson (email omitted because he never consented to me including it).
He ran the cable down the little hole by the passenger footwell, then up along the trim so that his player (an iRiver IHP120) can sit in the cupholder.

The cable is run under the center console trim panel.

The IHP120 can sit in the cupholder. Note that the console hides the cable very well.