I just had a similar issue and restored my forums in the following manner:
1. I copied my entire phpbb directory onto the new server (TotalChoice)
2. I created a new database and username
3. I gave ALL priveleges to the username for the database
4. I modified my config.php file to reflect the new dbhost, dbusername and dbpasswd and uploaded it to the server, overwriting my old config.php file
5. From cPanel I went into mysql account maintenace and loaded phpmyadmin
6. In phpmyadmin I clicked on "databases" (In menu "Show Processes" , "Databases" , "Export" , "Log out". I then clicked on my database name and then the "SQL" tab. I chose to load up a text file by clicking "Browse". This text file was the .sql backup generated by the built-in backup function in phpbb. (My old host did not have phpmyadmin so I was backing up my database using the phpbb backup function in the administration section of phpbb).
7. I connected to my new phpbb site! Everything seems to work great!