Diamonds are fairly easy to clean. Most can be cleaned with a mixture of 1 part ammonia and 6 parts waters. Soak the diamond in the mixture for a few minutes then brush off with a soft brush, a small artist's paint brush works well. Since diamonds can be set in different metals, be sure the mixture won't harm the metal. For example, silver shouldn't be exposed to ammonia. These pieces can be cleaned with a steam cleaner like the Scunci Steam Cleaner. Just hold the jewelry with a pair of pliers wrapped in microfiber and shoot it with the steam from all angles. Rinse in warm water and pat dry with a microfiber cloth. An ultrasonic cleaner filled will water will also work well for diamond jewelry, but be sure to check all settings before and after. The vibration can shake the diamonds loose.
You should have your diamond jewelry checked and cleaned by a professional jeweler at least once a year. They will be able to clean the stones and verify the security of the settings. Many jewelers will do this free of charge.