You can enable journaling of all messages that pass through Postfix just as you would with Exchange. Postfix 2.1 and later includes the “always_bcc” option, where you can enter an email address that receives a "blind carbon copy" of each message that is received by the Postfix mail system.

To enable this, simply edit your main.cf (location varies depending on your operating system) and add the following: always_bcc = email@address, where email@address is the same address you would use for exchange journaling. Restart Postfix to make the change active, or simply rebooting (#postfix stop && postfix start) will typically work, but see your vendor documentation for restarting postfix.