Invoke a command/script disconnected from the controlling terminal?

You need to start another session not attached to a terminal, so for instance:

$ setsid sh -c 'tty; ps -jp "$$"; echo test' < /dev/null > log 2>&1
$ cat log
not a tty
  PID  PGID   SID TTY          TIME CMD
19506 19506 19506 ?        00:00:00 sh
test

See also the start-stop-daemon command found on some Linux distributions. There's also a daemon command.