These are the steps I have used to update 4 eZ sites from 3.4 straight to 3.7.3.
Create database
Copy database from live site
Install eZ
Use eZ upgrade tool and phpMyAdmin to synchronize the old data to the new database structure
Manually set the "container" flag on the custom classes and folders for the system (there is probably a script for this)
Copy the templates from the live site, change ownership and permissions
Update override.ini
Update template priorities
Copy the stylesheets Integrate new CMS into existing site
The only odd thing was that the variable name $name didn't work well with 3.7.3 and I used $nname instead
Look at this topic: http://ez.no/community/forum/install_configuration/upgrade_from_older_versions_than_3_5_documentation_missing
Read carefully all upgrade documents. For 3.4 -> 3.5 yes you need to upgrade to 3.4.4 What scripts to run and in which order are described in upgrade docs.