I found the following procedure listed on a couple of forums and websites. Just tried it out with my iPhone 5S on a Windows 7 VM with a Ubuntu 14.04 host and it works fine.
Install VirtualBox and the VirtualBox Extension Pack. Find them both here. Create a Windows virtual machine.
On your host Ubuntu machine, run the following command -
sudo usermod -a -G vboxusers <USERNAME>
Replace <USERNAME> with your username, of course.
Log out and log back in. (This seems to be important!)
Run VirtualBox. Open the Settings to your Windows VM. Under the USB tab you'll be able to choose your Apple device by clicking on the + button on the right.
Run your Windows VM now and install iTunes. You'll be able to sync and backup like normal.
Hope this helps you out!