Where do you get the oxides?  Almost all calcium oxide is produced by baking the carbonate, which would tend to defeat the purpose.
That is the purpose, not to sequester the CO2 but to concentrate the CO2 from the few hundred parts per million in the atmosphere up to near 100% for sequestering elsewhere
in a cyclic process.
If you are going to capture CO2 for something, you might as well capture energy at the same time.

There's some research I just got the goods on (direct carbon fuel cells) which is pretty much ideally suited for a cycle which does just that.  I hope to be blogging that as soon as I get enough think-time to organize it.

One suggestion I have read about (and discussed with some expert aqueous geochemists) and thought about my little own self, as a Marine Geologist/Structural Geologist is that large amounts of MgO (which would react to form MgCO3, aka Dolomite), FeO (which would react to form FeCO3, aka siderite), MnO (to form MnCO3, or rhodocrosite)...and so on Here is a LONG list of potential candidates comprising the large and varied carbonate mineral class.

How to get those various oxides WITHOUT resorting to methods that release CO2?

Grind up mineral deposits.

Where to get those mineral deposits, specifically some of the more exotic ones?

Ophiolites and the ocean floor.

MnO, MgO, FeO and some of the other weirdo oxides are found in abundance in those geologic settings.