If too much rain was a problem, you may have a water drainage problem caused by such things as clayish soil or planting in a location where water accumulates. Planting elsewhere or in a raised bed may be a solution.

Cutting the roots vertically every inch or so will prevent the roots from circling and eventually causing damage. Azalea roots are tiny, fibruous roots and should be handled with care (or not handled at all). One way to tell that you are in the zone where the roots can be found is to assume you have roots up to the drip line.

If there are no roots circling or if you do not see many roots, do not necessarily worry as they may take a while to grow much, especially if there were moisture issues before. Assume a year but I do not recommend that you disturb/check the roots often.

I guess I did not phrase my statement well. The cuts I was going to make on the root ball would be 1/2" to 1" deep vertically, but only 4 or 5 cuts(total) so that the roots would grow outward rather than circling the ball. I would do this before planting. After planting I did not plan on checking the roots.
My question as to when I should expect to see roots coming from the root ball was in reference to the plant that died. When I pulled it from the ground there were no visible new roots after 5 mo.
The beds are raised and prepared with proper soil for acid loving plants. No clay soil. Good drainage.

Char, how vigorously you need to roughen up the rootball to loosen encircling roots will depend on just how potbound the plants are. With their network of fine roots, sometimes rhododendrons are sold in amazingly rootbound condition. If they are in loose potting medium and have been correctly potted up seasonally, you can do this with your fingers, or the cuts you describe. You can also wash away some of the container medium to free some roots with a hose. If the plants have extremely tight rootballs, it's sometimes necessary to cut away matted roots from the outside edges of the rootball with a sharp knife all the way around, 1/2" from one gal container plants to as much as an inch from those in larger pots.

Your earlier plants should have begun to establish at 5 months and not lifted out of the ground with their pot shape precisely intact.

Be careful when installing them not to plant any deeper than they were growing in their nursery pots. If you have recently prepared the soil in your raised bed, plant a little high initially to allow for settling, then mulch after planting - important in any case to cool soils and conserve moisture.

Char, when cutting the root ball, the cuts are on the sides of the root ball, not the top. They are about every inch or two around the outside.

One problem that even good landscapers frequently don't diagnose, is a plant that was root bound before it was transplanted into a larger pot in the nursery. When you plant it, all looks well. Take one of the dead plants and do a autopsy. Remove all the dirt. You will quickly see of the plant was root bound inside where you couldn't see it. Then cut several of the roots if they are brown inside, the plants died of root rot caused by wet, poorly drained conditions. If they are white inside, then it is something else.

The best indicator of nutrients is the color of the leaves. If they are a nice green, then the soil should be OK. If they are yellow with green veins, then they are chlorotic and it could be a nutrient problem such as iron, but is usually a pH problem, usually too alkaline.

Here is a link that might be useful: How to grow azaleas

Thank you all for your suggestions. The roots on the dead ones that I dug up this morning were "brown" not white. Looks like it was mainly a water problem. Absolutly no new roots showing in 5 mo. Checked soil Ph this AM- 6.5 I'll continue to try and bring this number down some with time.