If you can rule out the residue from some kind of spray or air pollution, then that leaves Powdery Mildew. This is a problem on some varieties of azaleas. It is worse this summer because we have had so much rain. It is worse on plants that are pruned heavily since they have very poor air circulation around the branches and stay damp longer.

A grayish white, powdery coating or fuzzy white growth on upper or lower surfaces is Powdery Mildew (Microsphaera azaleae). This is more prevalent on deciduous azaleas and sometimes it affects the lower surface (reverse side of the leaf) more. Entire leaves can be covered. In late summer and fall, small black specks may be found in the white areas. Powdery mildew is more severe on shaded plants. It is favored by the high humidity found in crowded plantings and damp locations. The disease is more severe during periods of cool, moist weather. These fungi produce spores on the surface of the infected leaves which are spread by wind currents to surrounding leaf tissue. These fungi over-winter in the bud scales for initiation of infection next season. There are a number of ways to manage this disease.

Do not over water or over fertilize plants, as the fungus prefers succulent new growth.

Hand-pick and destroy mildewed leaves to control small amounts of infection.

Hose diseased plants with water when practical. This can help remove fungus and prevent new infections.

Prune and space plantings to allow good air circulation. Do not plant in extremely shaded or damp areas.

Rake and destroy fallen leaves year-round to reduce infection source. Do not compost diseased materials.

Chemical control is possible. Begin applications when you first notice the disease on current-year leaves. If disease appears late summer, applications are not necessary on deciduous azaleas. Do not apply sulfur products when temperature is over 85F or within a few weeks of an oil spray.

Black Leaf Sulfur Dust
Bonide Lime Sulfur Spray
Bonide Remedy
Monterey Fungi-Fighter
Ortho RosePride Funginex
Ortho RosePride Orthenex
Safer Flower, Fruit & Vegetable Garden Fungicide
triadimefon (Bayleton, Strike)
Spectracide IMMUNOX

This fungal disease can weaken the plant. Spray when you first see the disease and then again in 10 days. Chemicals will not control the fungi that has already become established. For more information see the link below which includes a section on 'rhododendron powdery mildew'. The symptoms are different, but the organism and control are the same.

Here is a link that might be useful: Powdery Mildew