I'm running elementary OS (Luna). It mostly works fine, but I really need iTunes, which seems to only run on Windows and OSX. Is there any way to get iTunes running on Linux?
You can't without a Virtual Machine or Wine. Alternatively you could try Amarok and Banshee, which can also sync with your iDevice, and you can sort all music needs/etc with them. If you need to use the App Store you'll need to use the website AppShopper.
If you really need iTunes I think the only option is to run it with wine
Another alternative would be to spin up a Windows VM either with VirtualBox or VMWare workstation. However, you would need to have a legit Windows license key for this.