Arborvitae can die for a few reasons. One possibility is that they were not planted correctly and the circling roots or possibly twine left in place finally strangled the base. If you are able to dig one out and wash soil from the roots to see what is happening underground, that's a good idea. It can be one of the dead ones.

Other root problems can be disease or watering disruption. Is the irrigation working the same as prior years? I had a friend lose the end of her hedge when a sprinkler was plugged and she didn't realize it until too late.
Overcrowding might play a part as well. Measure and compare to recommended distances for mature plants. Often hedges are planted double-density for quick fill, then owners forget to remove the alternate plants as they grow.
This article from OSU Extension gives the basics of checking out a dead plant to get clues to what is happening. "A dead tree or shrub? Learn from a postmortem analysis" http://extension.oregonstate.edu/gardening/dead-tree-or-shrub-post-mortem-analysis
I look forward to hearing more about what you find.

Thank you for your reply.  This is helpful, but I'd like to clarify further.  Since I planted these trees, I'm confident that they were planted properly.  This tree death seems to have come on fairly rapidly.  The trees were green thriving and healthy until last year. Two or three trees looked brown last fall, and now you see them.  Does this history narrow down the possibilities at all?  

I have noticed a number of other dying arborvitae around town this year, and a couple of my neighbors had large conifers die.  Do you know of a disease that is running through the area and killing these trees?  

Is there evidence/knowledge of increased deaths of arborvitae and their relations resulting from the unusually hot weather last summer? 
  
Thank you again

One more question:  If I take in or send you samples from the remaining trees, would you be able to tell me if there is a disease affecting them?

Thanks!

Yes, it likely is a result of the hot dry years, and not a disease. The stressed trees will get diseased and attacked by insects also, but the base cause is still the drought stress. Another piece of the puzzle is that a conifer can be dead at the roots for quite awhile and the top doesn't show it (think of cut Christmas trees). The browning shows up suddenly, but the tree death wasn't sudden, nor sometimes even the same year as when the tree finally looks dead. How would an owner even know it was short water?
I was going to send you a link to an article about the drought stress but I found so many, you might like to look through them yourself. Do an internet search for "Oregon State University conifer stress drought". I found blog posts from forestry and extension from 2013, 14, 15, and 16. Here is one from just this May: http://blogs.oregonstate.edu/treetopics/2016/05/26/effects-drought-continue-valley/
This one lists ways to keep trees healthy: http://extension.oregonstate.edu/gardening/2015/05/drought-hits-douglas-fir-trees-hard
Many of the articles include links to more information.
Bottom line here is, I suppose, that you can replant the hedge unless you find evidence of soil-borne disease.