I've found that using vinegar mixed with warm water is one of the best natural cleaning methods. The aroma could overwhelm the bird(s) but if you rinse the cage well enough it should dissipate.
Also, lemons are a natural disinfectant so lemon juice mixed with water is also a great solution. Especially if you don't like the smell of vinegar. I've used both in the past to clean bird cages and had no problems.