I know its going to sound weird but toothpaste. It really does a good job on cleaning jewelry. Give it a nice scrub then rinse it off with warm water then dry I promise it'll sparkle like the sun.

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try dropping it in a small bowl of warm water and dish detergent next time you wash your dishes. (just don't forget you put it there.) its not as good as jewelery cleaner, but it works. and try using a soft bristle tooth brush to get in to the small spaces, they tend to collect lots of icky stuff like dirt and dead skin.
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Real gold and real diamonds are pretty hard to ruin. Try rubbing alcohol, or even dish detergent. Little brush to get into the crevices but not a hard brush as that may scratch the gold. It won't scratch the diamond. Or ammonia if you can stand the smell.
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I concur with Larissa, toothpaste & a gentle soft toothbrush, rub gently & rinse with cool water then use a good polishing cloth. It will sparkle beautifully. Make sure you put the plug in the sink, you wouldn't want to lose it down the hole.
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Wash it in warm water and fairy liquid then dry and polish with a soft clean cloth.
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Use lemon juice and sodium bicarbonate.

Brush it with a teeth-brush.

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gotta go with the toothpaste. I use it all the time on all my jewelry and it works great!
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nbo