No, I have never heard of an azalea blooming alternate years. However, what can happen is one of the following:

1) Pruning. I know this isn't your problem, but it is the most common problem, so I am including it to be complete. The buds are formed in late summer and early fall so pruning then or later is not advisable since it will remove flower buds. New leaf buds will form in the spring, but new flower buds won't form until the next year.

2) Fertilizing. Nitrogen promotes leaf and branch growth and discourages flower bud production. It can also force late season growth that gets killed or stunted by frost damage. Phosphorus promotes flower bud production and hardiness. Potassium is necessary for well being. Improper nutrients my be a problem that affects cold hardiness and flower bud set. A few things you can do are to fertilize with phosphorus (super-phosphate) per directions to increase hardiness and flower bud set. This can be done any time. Do not use nitrogen rich fertilizers as they may inhibit flower bud production and also reduce cold hardiness. Lawn fertilizers are notoriously high in nitrogen and should be kept away from flowering plants.

3) Weather. Cold weather can kill flower buds. Usually you see the brown buds in the spring. Cold spells in the fall or spring can damage buds that are not hardened off. Bud blast (blooming in fall or winter) uses up good buds which are then not available at the normal blooming time. Early frosts are especially destructive. Plants harden off progressively in the fall. If a frost occurs before hardening is complete, the frost can kill flower buds. Late frosts in the spring after dormancy is broken is also a problem. This is especially true of early bloomers that break dormancy very early. Cold climates may be too cold for many azaleas. Most azaleas have a low temperature at which the flower buds are damaged and will not produce flowers. It varies greatly from variety to variety and somewhat from season to season. Flower buds can also be damaged by cold, dry winds, particularly when warm winter weather is followed by a period of bitter cold.

4) Sun & Shade. Some azaleas need full sun to bloom and others can take fairly dense shade. In general, the more sun the more flower buds but also the greater exposure to damage from desiccation in summer or winter. More shade produces tall spindly foliage and less flowers.

5) Bud blast. Disease may attack the buds before they open. If it is a fungal infection, the dead bud will be covered with short hair-like structures.

6) Bud set. The buds could be foliage buds rather than flower buds. In this case check the previous section (1-4) about flower buds not setting.

7) Fall Blooming. Some causes of fall blooming include a dry summer and moist fall and/or warmer climates including a warm fall. For some reason, fall blooming is more common in Georgia according to Dr. Sandra McDonald. Plants which are not sufficiently hardened off or are exposed to unseasonable warm spells can start bloom prematurely. These blooms are seldom satisfactory and many times get frozen before opening fully. In any case, the seasonal bloom is lost. Avoid application of nitrogen after late spring, but phosphorus and potassium may be applied throughout the growing season. Application of nitrogen too late in the season can hinder hardening off in the fall.

8) Acidity. Measure the pH and acidify if necessary. Flowers of sulfur (powdered sulfur) or iron sulfate are the best chemicals to use to increase the acidity [lower the pH]. Do not use aluminum sulfate since aluminum salts build up in the soil and eventually becomes toxic to many plants including rhododendrons and azaleas. If soil is too acid, the symptoms can be the same. Very acidic soil can prevent the roots from taking up nutrients. In very acidic forest soil, an application of Dolomite and a light topdressing of mushroom manure in late spring is all they need. Sprinkle the lime on in late winter, very early spring. Don't overdo it - just a light sprinkle. If it is mid-spring, get the lime on right away so the azalea roots will be able to take up the soil nutrients in time for new growth. If you don't have rain, water it in well.

9) Protection. If the plants are wrapped in burlap during the winter, they may gain a few more degrees in hardiness.

10) Drought. When soil moisture is too low, the buds will not open. Watering will usually resolve this condition if detected soon enough. Also, summer drought with a moist fall can contribute to fall blooming.

11) Deer Damage. Deer and rabbits may eat many of the flower buds as they browse in the winter, particularly if the weather is harsh and other food is scarce.

Thank you rhodyman for such detail. I have also been told to mulch with pine needles. Lots of that around here.

Mulch is very important but won't affect blooming. It protects the azalea's shallow roots from summer heat, winter cold, and dry conditions. It will make the effects of drought less severe, so in that way it will improve bloom after droughts. Also, it will prevent the thought of cultivating around them. Since azaleas have shallow roots, they can not stand disturbance of the soil around them.

Good Luck!