How to back up iCloud Calendars/Reminders/Contacts

No API. No protocol.

But there are three workarounds I can think of. The first is to use your mac and copy the following files manually (or rsync in combination with cron or something else you like).

• Mail data and settings ~/Library/Mail/V2
• Contacts data and settings ~/Library/Application Support/AddressBook
• Calendar data and settings ~/Library/Calendars/
• Safari data and settings ~/Library/Safari/
• Keychain data and settings ~/Library/Keychains/

(for sake, just copy the whole 'Library' folder ;) (list taken from http://support.apple.com/kb/PH13985)

An other way to do it is to setup your linux backup machine to fetch the mail, contacts and calendar and backup that like you normally would.

• email through IMAP (http://support.apple.com/kb/HT4864)
• calendar through WebDAV (it is likely that this must be a public url)
• contact, no clue. For android it is possible but I have not seen a Linux solution.

Yet another way is to install 'iCloud Control Panel' through Wine and handle your files from there, although it might work like garbage (https://appdb.winehq.org/objectManager.php?sClass=application&iId=14545)

A virtual machine running OS X on your linux server might also do. But I do not know if it is legit to talk about that on stack exchange.

Edit: see comments for ical and contacts