So, even though I could not procure any excuses by writing about it, I bought some new hardware for a desktop machine - an Asus M2A-VM HDMI motherboard, a 45W processor, a low energy harddisk and some RAM.
When the post office finally found the package they had misplaced, I could connect everything yesterday evening. Everything seemed to work great, after I downloaded the amd64 installer for the amd64 iso and not the i386 installer, until I started the graphical window system, X.
Every so often the screen would blank for 1-2 seconds - sometimes every 10 seconds, sometimes farther apart. I tried flicking on and off all the different things I could think of in the BIOS to no avail. So in the middle of the night, I decided to upgrade to the newest BIOS.
The flashing seemed to work fine, except after the message about the box rebooting in 5 seconds, the screen went blank, and nothing more happened. After pacing around a while, I decided to do the Clear RTC RAM dance (removing the battery, flicking a jumper and waiting a while). That helped. Phew.
But the blinking continued. I decided to run memtest86+ while I slept.
When I woke up, I started playing with the X.org configuration, twiddling configuration options here and there. Turning off acceleration made everything slow, but the blinking persisted. I tried upgrading to the X.org in experimental, that did not help either.
I tried asking the folks on #linux.dk on OFTC for help, and they suggested faulty hardware or an IRQ-conflict (that sounds so retro!) I wasn't able to figure out how to reassign the IRQs, but it was suggested that I tried the proprietary ATI-drivers. So I did.
The proprietary ATI-drivers simply blanked the screen and froze the machine cold. I was expecting them to have problems, but this really was impressive.
While mucking about with restoring X to the version in unstable and removing the useless proprietary drivers, I happened to notice that there is a free software driver called "radeonhd" as well as the "radeon" one I had started with. So I decided to try "radeonhd".
So far, no blinking.
And. It. Only. Took. 24h+.