Since my last answer was completely wrong, I did some reading on my own. And here is the solution you need:
Run the following command on your terminal:
# blkid
This will output the UUIDs for each device on your system. For my system, the output looked something like:
/dev/sda1: LABEL="boot" UUID="aa84c5a8-6408-4952-b577-578f2a67af86" TYPE="ext2"
/dev/sda2: LABEL="root" UUID="a430e0ef-fd35-432f-8b9a-75a49b89ad8a" TYPE="ext4"
/dev/sda3: LABEL="swap" UUID="e388806a-dc27-4f4e-a136-3d1ff4e53962" TYPE="swap"
/dev/sda4: UUID="088E027A8E026114" TYPE="ntfs"
/dev/mspblk0p1: SEC_TYPE="msdos" TYPE="vfat"
Next, open /etc/fstab in a text editor. Change the /dev/sdXY entries to UUID= and input he UUID's you got via the blkid command.
Now, I am assuming you are using the default grub2 config files. If you have manually edited them, I am sure you know what the edits are and how to make them again.
Run
# update-grub
Make any customization changes you want to your grub.cfg
Also, in some cases, you may have to update your initramfs with:
# update-initramfs -u -k all