Bluetooth keyboard and mouse connectivity problems

I have an Apple bluetooth keyboard and Mighty Mouse connected to my Macbook Pro 13", running ML 10.8.1.

The keyboard and mouse will constantly disconnect and re-connect on their own every few minutes. I'll see the "connection lost" notification, and then shortly after the "connected" notification. Also, even when the keyboard is connected, it often becomes unresponsive and/or strange things happen, such as the special characters menu appearing above the cursor (which I believe there is a keyboard shortcut to invoke, but I'm not doing that; it appears on its own).

This never happened in Lion.

Any idea what's happening and how to fix it?