A creaking noise that you are describing could be coming from the ice maker module if your model has an automatic ice maker. I would check this area first. If a batch of ice gets caught in the teeth during the ejection process, the ice maker module will make a constant creaking noise as it continues to try to eject the cubes. Freeing up this type of jam would eliminate the creaking noise.
This is the only source of a constant creaking noise that I am aware of that would be constant. This type of problem would not affect the proper cooling of the refrigerator or freezer.
If you need more help with this problem, resubmit your question with additional details including the model number of your refrigerator.