I think certain hygiene standards should be continued. A lot of people take it to extremes, and I believe a child should not only be allowed, but also *encouraged* to play outside, pet animals, roll in mud puddles, you name it.
Certain dishes however are, or can possibly be, contaminated with pathogens I wouldn't risk not eradicating. Say, salmonella from eggs/raw meat; assortment of parasites; or uncle's touch of a sweaty mustache. While claims like 'they didn't have dishwashers before and survived' are expected, *they* also produced their food in different environments, but had a lot more diseases to fight with as well.
The development of asthma or eczema mentioned in the study may not necessarily come from decreased dish-present pathogen exposure. Modern dishwashing detergents and rinse aids are loaded with a variety of synthetic compounds. Together with lack of proper rinsing current dishwashers are known for it could be the reason. My eczema disappeared completely since I switched to 'natural' products. A visit at a friend's house always brings it back with wonders of Febreeze and other diamonds.
Another thing, one should assume dishes alone don't hold nearly enough responsibility for exposing children to pathogens. A single sneeze at school would probably play a much more significant role than all that the kid eats from. Immune system works based on general, environmental exposure, rather than one particular source, obviously. Single studies like the one mentioned always drive me insane. If the diswasher-friendly parents ran their machine on Sanitize, there's a good chance they also sterilized childrens' bottles each refill, vacuumed 5 times a day with a True HEPA-equipped device, never let the kids play in common (or considered unhygienic) areas, and so on.