A few days ago I upgraded my computer with some new and used components, basically rebuilding it. Since then the computer has been acting strangely. In particular it restarts itself about once an hour, without giving any error message, and when I download files they usually end up as corrupted and failing CRC checks. Installing programs from CD's seems to work fine, but files downloaded from the internet or the LAN, through a Wi-Fi USB adapter or the motherboard's ethernet port and from different router/modem/ISP/line combinations are mostly unusable. The temperatures seem fine, never rising above 50c or so, and the HD's also don't show any errors through standard disk checks. The current configuration is a new motherboard (Asus P5VD2-VM), new RAM (Kingston 1GB CL5 667Mhz) and new CPU (Pentium D 925) two used HD's (WD 120GB IDEx2), used Leadtek PX6200 128MB gfx card, 400W generic PSU and running XP SP2.
Does anyone here have any idea what the problem(s) might be?

Could be almost anything - I'd suggest first checking the hard drives for problems. The free utility available here ( http://www.hdtune.com/ ) can give you an idea of the hard drive's status within Windows. I prefer the W-D diagnostics myself tho - you can get versions that will work within Windows and versions that you can boot from.