When you say the area is quite narrow, can you describe what makes it narrow? Adjacent buildings, sidewalks, driveways, etc.? Are roots constrained by such structures? How many hours of direct sun does this area receive during the growing season? What is the soil like? Good garden soil; slowly-draining, heavy clay; quickly-draining sandy or gravelly soil? These features will impact your options.

You might want to have your soil tested. Arborvitaes thrive here in our deep, acidic soil. I'm not sure they do as well in alkaline soil, especially if other factors are involved. Are they near a wall or walk which might leach lime into the soil?

As the above poster says: your options will be restricted by your growing conditions.

Looks like my last post didn't go through... sorry.

The shrubs are planted on a short berm (a foot high). On one side is a narrow sod walkway and on the other, our neighbors lawn. Nothing that might leech lime into the soil.

The roots are not constrained but the soil is heavy clay.

Half of the shrubs are shaded by the house and receive full sun from around noon until sunset while the other half get full sun.

Sunlight doesn't sound like a problem, and the roots aren't physically constrained by concrete. Soil conditions for the roots might either be too wet or too dry. Might be time to test soil drainage: (1) dig a hole in the berm about as deep and wide as the original rootball of your arborvitae (or use one of the holes left when you pulled out a dead one); (2) fill the hole with water and let it drain out, to wet the soil; then (3) again fill the hole with water, but this time time how long it takes to drain away. If the water drains from the hole in less than 4 hours, then drainage is probably OK. If the water drains away so quickly that you have trouble even filling the hole, then drainage might be too good, and the soil might be drying out too fast for the roots to survive. If the water takes 6 or more hours to drain away, then the drainage could be poor enough that the roots might be suffocating (they need oxygen that diffuses into soil moisture from the air) by drowning.

Could something sprayed on the neighbours lawn affect the arbor vitae?

Just to be clear, the whole shrub has not died yet. Sections of it are slowly turning brown and dying little by little. It's almost as if only sections of the roots have been affected. Other parts of the shrub look perfectly healthy... for now. This is happening to about 5 of them some more or less than others.

Have you stuck your fingers in the soil around your arborvitae periodically to know what the soil moisture conditions are? It really does sound like a root problem. If the berm is very narrow, soil in the berm could be drying too quickly. Otherwise, it's likely too wet. The only way to know for sure is to to look at the condition of the roots (healthy roots should be firm and have tan-brown outer covering, with creamy-white interior; rotting roots would be more uniformly brown and perhaps very soft) and to investigate the soil drainage. Arborvitae can tolerate periodic soil flooding, but not lengthy, prolonged wetness. I have seen some instances of Techny and Emerald Green arborvitae planted in low berms over wet, heavy clay soils in the Chicago area appear quite healthy for 1-2 years and then fail over the next couple of years as their root systems extend beyond the berm into the surrounding wet clay. Do you live in a subdivision built after the 1960s? If so, then poorly-drained, compacted clay soil may likely be the problem.

My son works with trees in a nursery in the midwest and he said Emerald Green Arborvitae are not doing as well as they expected them to do in the midwest here. Sounds like for all the reasons listed above. If it's not the heavy clay and alkaline soil we are notorious for, they will winterburn easily from winter winds, at least when young. Junipers do well here.