I have a Logitech Music Anywhere USB transmitter that sends the sound to my stereo, and since I'm only interested in sending the Amarok sound I have chosen this simple solution: I identify the sound cards by running cat /proc/asound/cards and get the following result: 
0 [SB            ]: HDA-Intel - HDA ATI SB
                      HDA ATI SB at 0xfe020000 irq 185
1 [tra            ]: USB-Audio - Logitech Music Anywhere USB Tra
                      Logitech Logitech Music Anywhere USB Tra at usb-0000:00:13.0-1, full speed

Then I have just set the mono and stereo ALSA device of Amarok to plughw:Tra
This solution works perfectly on my Asus Eee PC running the pre-configured Asus version of the Xandros Linux distribution based on Debian Etch. I tried the same on my desktop PC where I run Debian Etch and when playing Amarok it is clear that the sound is more or less sent to the USB transmitter since the blue communication LED goes on (both on the transmitter and at the stereo) but no sound comes out from the stereo. I have tried to find all mutes and volume controls but I suspect that there is a problem more related to the hardware driver. The reason for believing that is that I can actually probe the transmission from the USB device indirectly by placing my computer speaker very close to it and the interference is much more pronounced from the working Eee than from the desktop PC. This interference is not dependent on the sound volume from the Eee PC or Windows on the desktop PC and even remains the same if I chose to mute the sound, indicating that the sound level is not my problem. The USB device works perfectly on Windows on the desktop PC, so probably no pure hardware problem. Does anyone have any good idea about how to proceed? I thought of comparing the actual hardware sound card drivers on the Eee PC and the desktop but I have no clue how to find out what driver I am using. Maybe someone could guide me on this?

Thanks!

Maybe running 'lsmod' helps you. The sound card drivers are prefixed snd_*
Also, you can have a look into /proc/asound/card1 (in your case). There might be more valuable information for you there.
Furthermore, looking into /var/log/syslog or /var/log/dmesg can give you hints what driver is used for this device.

Thanks marcel for some great pieces of information! When originally posting this message I really hesitated to do it on the Amarok Forum since might not have much to do with Amarok. Maybe it hasn't, but I have now by marcel's hints checked that both computers seem more or less identical and in addition I have made a test where I come to a strange conclusion:

Having understood a bit (just a bit) more about /dev/dsp I have now made a real weird observation on my Eee PC where my USB transmitter works perfectly from Amarok. If I send any file to /dev/dsp (i.e. cat file > /dev/dsp) I get some noise out of the Eee speakers as expected. I guess that /dev/dsp is as close as I could get to the hardware? Ok, since Amarok also works on the USB, i.e. /dev/dsp1, I tried sending the file to that one instead, "cat file > /dev/dsp", and surprisingly I hear no noise! However, it triggers the status LED of the USB tranmitter and causes interference with the loud speaker nearby, e.g. reacting exactly as my Debian Etch PC when running Amarok. The conclusion would then maybe be that Amarok does not send its sound to /dev/dsp1 since sending data there doesn't bring any sound from the USB tranmitter to the stereo. But this is probably a wrong conclusion since when I have Amarok running with great sound out of the stereo via the USB transmitter and try to send a file to the /dev/dsp1 I get the error message that the device is busy. So, Amarok manages to use /dev/dsp1 to tranmit sound over the USB whereas this isn't possible when just sending a file. So, does Amarok on my Eee works differently than Amarok on my Debian Etch? I have the same configuration in Amarok, but I am using 1.4.3 on the Eee and 1.4.4 on Debian.

I am relatively confused and am now desperate enough to throw out Debian Etch and install Ubuntu just to try something completely different!

Solved by installing Ubuntu on my desktop PC. It is a pity since I really liked Debian. So if anyone could come up with an explanation why this could happen I'll be happy.

How stupid can I be? When trying out everything I was testing a .wav file on the Debian system while testing mp3's in the other cases! Amarok now works well on the Debian system also but there is then one problem to solve: With the internal soundcard of the computer I am able to play .wav as well as mp3 with Amarok but with my USB sound card I can play mp3 but not .wav? Strange or expected?

Just passing files to /dev/dsp is not a good idea. It works for wav but should not work for mp3 (someone has to decode the file first and I am not aware that alsa does this for you).

Make sure that your configuration for your USB thingy is correct and that you have an up-to-date driver running.

Use your favorite player software and configure it to use your USB soundcard and test it.