Turnips and beets are roots too.
http://www.bonappetit.com/trends/article/what-the-irish-ate-...
I don't doubt the importance of dairy but I doubt it accounted for the majority of people's calorie intake.
http://www.ravensgard.org/prdunham/irishfood.html
The TLDR of their diet is "mostly pottage and soups, heavy on meat and dairy. Some oats and barley were grown, but a lot of that ended up being made into alcoholic beverages."