I've started having the same problems with my forum too, also recently upgraded from phpBB2 to phpBB3.
I usually use Firefox and this is where I first spotted the problem, IE7 is doing the same thing too. No amount of clearing of cookies seems to be making any difference. Any user who logs in can perhaps click on a forum or subject before being kicked out. I managed to get as far as posting a new message but every time I submit I get asked for username and password again.
Even disabling things like session IP validation has made no difference. Any help would be appreciated as my board is currently unusable.
Here's the vital information:
Your board's URL: http://www.eplicehockey.net
Version of phpBB3: 3.0.0
Was this a fresh install or a(n) update/upgrade/conversion (please be specific)? Upgrade from 2.0.22
Was this an install through your host? Installed on my own server
MODs you have installed: Ajax Shoutbox
When the problem started: In the last few hours