Here's what I did:
downloaded the latest mainline kernel parts from http://kernel.ubuntu.com/~kernel-ppa/mainline/ to my /Downloads folder
3 files (since I have a 64-bit system, I got the amd64 ones):
- generic headers for amd64: linux-headers-3.12.0-031200-generic_3.12.0-031200.201311031935_amd64.deb
- headers for all: linux-headers-3.12.0-031200_3.12.0-031200.201311031935_all.deb
- image for amd64: linux-image-3.12.0-031200-generic_3.12.0-031200.201311031935_amd64.deb
I tried to install them by just clicking on their icons in the /Downloads folder, but my default package installer (Ubuntu Package Installer) balked, saying something about missing dependencies.
So I did the install a different way:
In the terminal I set the active directory to the /Downloads folder, which had no other files in it. Then I opened the packages, and finally updated the grub and rebooted:
As the packages were opened and installed, I did get some error messages about some missing NIC firmware:
sudo dpkg -i *.deb
sudo reboot now
"possible missing firmware /lib/firmware/rtl8168g-3.fw for module r8169" (for example)
But other than that, everything seems to have gone smoothly.
(Haven't tested extensively, but the bootup was smooth and no glaring problems as yet. My internet connection is working fine, despite the NIC firmware warning.)
Thanks to everyone for the replies and suggestions.