Quite a number of things. These two gadgets don't work which is a pain because I use them all the time, every day:
http://www.dailysnooze-gadgets.com/ There is a workaround for 64 bit but it will only work with Vista.
My Plantronics headset which is totally reliant on Windows drivers won't work correctly, the mike is muted on some applications and I'm having problems getting other devices to work such as a HP laser printer.
I have a few older applications that are doing strange things such as Photoshop 7.0 which continually crashes when I try to save a files of a certain size (in terms of x,y dimensions).
The major one though is my Asus motherboard which is constantly warning me that my processor is overheating. It isn't but it thinks it is. This particular motherboard usually has an application running in the background that controls BIOS functions at an OS level. That won't run at all and I'm assuming that's causing the issue but I've not really done any research on it.
I only have 4GB of RAM so it's not essential that I run in 64 bit and it's no big deal to go back to 32 bit.