phpbb3 does work fine on a php5 server, so lets for a sec its not the problem, (but still can be) when you mean a new server, you transfered the DB across,, backed up and restored? have you tried with a fresh install? as its a bit pointless backing up the DB at this stage when its easier to start again,
its possible that either the backup or restore might have failed, are you using the same database version? what version of phpbb3 are you using? or are you using phpbb2 in which case you have posted in the wrong forum, (known to happen)
Yes, I export the database (by phpMyAdmin, in a text file). Then, on the new server, i've upload the forum files (the same as on the old server).
I use another database version, but that must be no problem I think.
I use phpBB3 Beta4.
Sorry for my bad english, I am Dutch.....