Vinegar would be a good place to start...if should dissolve the build up, or at least soften it up with bit so it can be scrubbed out with a kitchen sponge with a green scrubie.
Yes, just soak papertowels and place on the item. Leave sit for a few hours and then wipe off.
Thanks for the answers, but I'm still hoping for more ideas.
CLR is made to remove calcium, lime, and rust (hence the name.) Follow directions and wear good rubber gloves. I would use a cloth rag and soak it with the CLR and leave it sit on the areas you want to clean for a while. Then, scrub brush or scrubbie to clean away the buildup. I've never had any material react badly but to be safe, try it on a small spot away from the most visible area...
At the grocery store where the spices are located you can find Citric Acid for home canning tomatoes. A teaspoon of this in a 1/2 gallon of water would be a good place to start. I usually soak things that are way corroded in a stronger solution. You will be amazed what this will clean up! I really like the fact that it is doesn't smell and I don't need to wear gloves for protection. This can clean up corroded rusty metal or ceramic. I would try it on other things as well.
You could try using vinegar and baking soda. It will break down the calcium.