MySQL v5 is brand new (literally, it was released just in the last few weeks), so I'm not suprised that phpBB doesn't support it yet.
It would probably work fine; it's just that phpBB is checking and finding a version it doesn't know about, and preventing you from using it.
MySQL 5 has a lot of new features (some of them have been long-awaited), but phpBB won't use any of them, so my first suggestion would be to switch to MySQL v4 if you can.
If you can't (or don't want to) downgrade your database, your best bet would be to ask on the forums at www.phpBB.com
whether they're planning to release an update to support v5 (I assume they will do fairly soon). And there's bound to be other people there with the same problem, so they may have a solution already.
Alternatively, if you know enough about PHP, you could try tweaking the phpBB code to accept MySQL v5 as if it was v4, as it should all still work the same once you've got past the version check. But that's something I'd prefer to avoid, at least until you've spoken to the guys at phpbb.com.
Hope that helps.