Open fermentation was one. Using fruit skins/raisins, like you mentioned. Yeast sticks were common; at the end of fermentation brewers would dip sticks in the trub, hang the sticks to dry, and on brew day dip those into the wort. They would store them and re-use them. I also wonder if there was yeast present in the Bappir bread that was used in some early brewing. Some yeast surviving the baking process and fermenting the wort, possibly on the dates and fruit in the bread itself.
You might find this interesting.
The Yeast of the Ancients
(It's kinda new agey, but has some interesting info that could provide hours of interesting googling.)
Why brewing ancient Egyptian-style beer is so popular & how to make it
The maltose falcons had a great article on brewing with Bappir, but it looks like they're redoing their site and the old links don't work.
Found it http://archive.maltosefalcons.com/recipes/20051002.php
I shoudl have given you enough to further your google adventures.