You can run Premiere Elements 3 on Windows 8, but if Quicktime and the Avid codecs are installed, then Premiere Elements 3 fails to open.
I couldn't figure out why Quicktime was preventing PRE3 from opening. As I was looking through the Quicktime folders, I saw that I had installed Avid codecs. After uninstalling the Avid codecs, PRE3 started.
If you are wondering why I installed PRE3 on my Windows 8, I was trying to answer a question about PRE3 from someone using PRE3.