Finally managed to get Omeka installed on a test Ubuntu server. Able to log in and create a new Item Type and add a test record.
Could you try the steps to display error messages, including error logging, and try again? Then look at the end of the
application/logs/errors.log file to see if new messages are being logged.
That info should help us track down what's happening.
Problem is resolved. Turns out the "RewriteBase" in .htaccess needed to be set.
Ah! Easier than I thought, then!