Don't tell your staff that a customer is 'moaning'.
Log it. Keep a record, not just a Post-it. The business looks bad if someone follows up a complaint and you have no idea what they're referring to.
Speak to them. If they have written in, ring them back. It's much easier to build up trust if there's human interaction, and the customer will know that you are on the case.
Deal with it. Don't put a problem to one side and hope it will go away. Looking into complaints may not be the fun part of your job, but delays will just make the situation worse.
Give a timescale. It's much more reassuring and professional for the customer to know exactly when you will get in touch. 'Later this afternoon' will sound like you're trying to fob them off.
Be fair. Make sure employees aren't just made scapegoats for misunderstandings. Launching into a tirade against a junior member of staff to save face will make everyone hate you.
Complaints always arise. With all the care and attention in the world things can still go wrong. But if the same complaints keep coming in, warning bells should ring. How a business responds reveals its character.