I actually did this for one site. The Windows version of CS was 7.8.1.16, so it was using the PostgreSQL database. I just did a database backup on the windows machine, then copied it to the Linux machine. Then the longest part - copying the recording files from the Windows to Linux machine. After the files finished copying, I set the correct permissions on the recording data folder, then restored the backup, restarted the machine and everything worked. It's been running for 8 months now with no issue.
WinSCP and putty were indispensable.