A lot depends on when the tree is felled. If a tree is cut in winter, it is possible that the root system will experience an additional year of growth. What happens to the root system depends on many variables, including soil type and presence of other trees of the same species adjacent to the one being felled. If the same species is present and the root systems of neighboring trees are grafted, the roots may be kept alive by trees still standing. Some species, like aspen, will actually sprout new shoots along the root system. Often a stump with roots that are alive will seal the cambium on the stump surface, resulting in a round bump along the entire circumference of the tree right over the cambial area (i.e. just inside the bark).
The first question to ask is what type of irrigation is being used currently on the hedge. Most landscape oak root fungus problems are due to overwatering in summer months. However, Thuja orientalis (=Platycladus orientalis) is quite susceptible to Armillaria even without heavy irrigation as highly susceptible species do not require excessively high soil moisture to be killed.
Prior to any new planting, as much of the old hedge (including roots) should be removed. In addition, any other decaying wood should be removed. Replanting with plants that do not require supplemental watering is best. Some CA natives that make good hedges include: ceanothus, island mountain mahogany, manzanita, and toyon. Non-native possibilities include Leyland cypress and Japanese privet. Keep in mind though that even plant species listed as resistant to oak root fungus will die if overwatered.