With thanks to slackbuilds.org and Heinz Wiesinger:

jackd has to run with realtime privileges.
One way to do this on Slackware would be to use set_rlimits (not part of
Slackware). Since 12.2 there's another way, though experimental.
If you have a filesystem that supports posix capabilities (reiserfs does not),
you can grant jackd the rights to run in realtime mode, even when started as
normal user with the following command:

setcap cap_ipc_lock,cap_sys_nice=ep /usr/bin/jackd