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   To be absolutely positive as to the cause of your brown spots on your rhododendron leaves a sample can be sent to the Plant Diagnostic Lab using this webpage
Submit Samples - Plant & Pest Diagnostics (msu.edu)

  Brown spots can be caused by a variety of issues, but the most common is one of many forms of leaf spot from either a fungus or bacteria. The more common fungal leaf spot causes are from either an Alternaria or Septoria fungus. These can be addressed with a sulfur-based or copper-based broad-spectrum fungicide to prevent further spread. Prune off the damaged leaves prior to treatment to ensure spores cannot shift to uninfected leaves. 
  Bacterial leaf spots are harder to treat. Watering at the base of the plant will ensure healthy plants. Inspect periodically and remove any branch that has a cut in it - a place where bacteria can enter.  A copper-based bactericide 
may help in managing the disease. 
  In both cases stressed plants are more prone to disease. Our Michigan weather can play a role here as the high humidity is a major factor in spread of these leaf spot diseases. There are some practices that will lessen the amount of disease and can be implemented rather quickly.

    Remove all the leaves and stems that are affected as well as those that have fallen on the ground and throw them in the trash - do not compost them. Disinfect (10% bleach solution or rubbing alcohol) your tools between cuts to avoid the pathogen to spread.

    Prune to maximize air circulation within the plant.

    If and when you water, avoid overhead watering and choose a drip irrigation that waters at ground level. 

   Choose a balanced fertilizer for acid-loving plants to promote healthy growth. Excess nitrogen promotes lush growth making the plant more susceptible to disease.

    As a last resort apply a fungicide or bactericides following all label instructions carefully.

For more information:
Azalea and Rhododendron Diseases (psu.edu)