You have probably watched movies where someone takes a diamond and uses it to scratch a mirror or, cut a hole in a window. If you have diamonds, the first thing you should do is keep then separate from other jewelry as they can, and will scratch. If your diamond didn't come in a box, find some soft cloth to wrap it in when you are not wearing it.
The second thing you should consider is always having your diamonds professionally cleaned. This way, the jeweler can check for loose stones and fittings at the same time as they are cleaned. If you wish to do your own cleaning, there are products you can buy that will clean your diamonds and other jewelry using ultrasound. These small machines are quite cheap and seem to do a good job.
If you don't want to buy an ultra-sonic cleaner, you can simply wash you diamonds in soapy water. A tooth brush is the perfect size for cleaning the back of the diamonds as the bristles can easily clean in-between the fittings around the diamond.
Finally, yet importantly, there are many commercial diamond cleaning products available on the market. Do take care though if you are cleaning your own diamonds. Make sure the product you are using is suitable and will not damage your diamond or fitting.