This method is applicable for any diamond ring as well as many gemstones. However do NOT use this method with emeralds or any heavily included gemstones (with a lot of fractures / internal flaws) or porous material such as turquoise, lapis lazuli or amber. You can use a very mild soapy water or just plain water to soak and soften dirt particles and be very gentle with the brush!
Before doing anything check that the diamond or gemstone is secure and not loos in the setting. If it anything is loose take it to your jeweller to get checked and tightened.
If in doubt I would recommend you visit your jeweller and ask them to check and clean your jewellery for you.
I use a glass container with enough Methylated spirit to cover the entire ring and let it soak for a little while. If the ring is particularly dirty you can leave it overnight to soften anything that has stuck to the bottom of the diamond / gemstone and around the underside of the ring.
Here is my peridot and diamond ring soaking. After some time soaking you can take it out and gently brush off any dirt from the ring using an old tooth brush. If it hasn’t been cleaned for a long time it may take a few times before it is back to its sparkly best.
I always recommend to all my clients to get their jewellery checked every year for wear and tear and general clean.