Urgh, I tried starting my old comp today to transfer the most important docs to my laptop and put the HDD's on share for nino's method..
I failed. It starts up, it shows the Acer-screen with the option to enter Setup menu or boot menu, but it freezes there and none of the keys works, so I can't enter anything..

Anyway, I just bought an USB to IDE/Sata adapter, which, if it does what I think it does should allow me to access a HDD externally (that is, without having to build it into the new tower.. and thus save the warranty and possible damage from noob-skills). Let's see.