[zappsan]

Well, it depends on which host it is.
Some hosts will transfer your site for under certain conditions.

I had CPanel on my old host and it also came with the new host I signed up with (A Small Orange).
I simply contacted them and they moved everything for me.

I've also moved hosts before, without help from my hoster and didn't have any problems doing so.

I used MySQLDumper for the backup of my database and the restore on the new location and everything worked fine.
It might also have some advantages that my forums are rather small, moving bigger forums might probably be a little more complicated.

I'd just try contacting your new host, asking them how much they can help you.