I want to sell my desktop PC, and I will keep the hard disk so no one can recover my personal files, passwords, history or anything. But I was wondering if anyone can recover any private data like passwords or key-files from the RAM or motherboard? And if it is possible, how can I clean my RAM (or should I keep that, too)?
There's nothing that should be stored in the motherboard or the RAM that's of any value at all. I've not seen any forensics tools that can recover things off switched off RAM (the cold play attack wouldn't work on a system that hasn't had its RAM frozen in situ), or from BIOS.
Both recovery scenarios you're talking about are not viable and it's safe to sell on your RAM.
If you want to be "sure", make sure any motherboard passwords you set have been cleared, and/or reset the BIOS. Remove the memory modules, then put them back in place. This works since the RAM won't have anything stored once it's switched off.
That way you'd clear any information that probably isn't there anyway.
Journeyman's answer is all you need for RAM, but I don't think you need to worry about selling on a HD either. For modern disks, even a single pass with zeros or random bits will make data effectively unrecoverable by anyone without some pretty serious equipment. See here for greater detail.