Since upgrading to Mountain Lion, a lot of people have been having issues with their Time Machine backups that worked perfectly fine on Lion. Here are a few tips to try and solve the problem:

Step 1) Shutdown your computer and restart, sometimes the Time Machine drive can have an issue and it won't be able to fix itself until after a reboot. Try this first.

Step 2) Remove the drive through System Preferences > Time Machine and then readd the device. As long as you don't wipe the device of all its data, your save points will still be there.

Step 3) Run Repair Permissions through Disk Utility on the Time Machine device. You can also try running Disk Repair, also under Disk Utility.

Step 4) Under System Preferences > Energy Saver > disable the "Put hard disks to sleep when possible" box. This option has fixed the issue for most people.

Step 5) If none of these steps worked for you, the only thing left I can recommend is a reformat and set the disk up as new again.

Let me know if there are any other things you have tried to do to get your Time Machine working again. These issues shouldn't be happening, but the fact of the matter is that they are. If you have any questions, feel free to leave them below in the comments box!