Prior to connecting a MySQL Database, follow the platform specific instructions on whitelisting Periscope's IPs.
MySQL databases identify users by the username and IP the user is accessing the database from. As such, if there is a certain user database, whitelist IPs within MySQL by doing:
GRANT SELECT ON my_database.* TO 'periscope'@'%';
In place of the '%' which is a wildcard character that allows connections for the periscope user from anywhere, the user can instead substitute for the two Periscope IPs, or the IP of of the SSH box if Periscope is accessing the MySQL database through an SSH tunnel.
Finally, follow the instructions on granting table permissions so that Periscope can query tables.