About 3 weeks ago, we had fifteen 7' American Arborvitaes planted. We have had plenty of rain and we have watered very regularly. Most of them are starting to show brown. They do not necessarily seem to be browing from the inside out. I'm wondering if they may be getting over-watered. Our soil has a lot of clay in it, so I'm also thinking the water may not be draining effectively when watered. Any advice on watering practices to help before this gets any worse?

It is possible they are getting too much water considering what you indicate - plenty of rain, additional water, a clay soil, and possibly poor drainage. It would be easy to see how effective water drains away from the site they are planted. The general rule of thumb is to water 1 inch per week when rainfall is lacking. The watering needs to be deep enough - it needs to be thorough, not a drive by with the hose. You can determine how much rain has fallen and determine how much additional moisture is needed. Waterlogged soils are never good as this encourages root rots Transplant shock is another possibility. You could always dig now and see exactly how wet the soil is. It is not uncommon for some yellowing or browning to occur. Depending on the size of the plant vs the size of the rootball, it could be going through some stress. You indicate that you had them planted. If this was done by a landscaper or garden center, you might want to check with them if the condition persists.