Evidently there are naturally occurring minerals that absorb carbon dioxide from the atmosphere. These minerals are abundant. There may be enough in the United States, for example, to store 500 years' worth of US carbon dioxide emissions. But the rock absorbs CO2 slowly. It would take thousands of years to absorb an amount needed to have a significant impact on global warming. Trying to speed up the process is expensive and in the lone demonstration project cited in the article, the process consumed so much energy that it created more carbon dioxide than it absorbed. More in a recent news article here.
The article makes an argument for more research funding on mineral sequestration:
Insufficient research funding is the working scientist's perennial complaint. But considering how much money the federal government has already extended toward dubious climate solutions like biofuel, and considering how overwhelming the need to develop big solutions to climate change has become, it's difficult to understand why mineral sequestration — the potential merits of which are so impressive — hasn't garnered more attention.
The article doesn't address the costs of mining the material. How much digging would it take to mine a significant amount, and what would be the environmental consequences of the mining? Here is an older academic paper that raises some of these issues.
Here's another short summary with nice graphics that identifies these other issues as well, from about 2002. An interesting bit of overoptimism: this paper cites a prediction that mineral sequestration will be successfully demonstrated at a cost of one cent per kilowatt-hour by 2007. (Average cost per kwh in the United States currently is between 10 and 11 cents.)
Still, it doesn't seem we'd lose much if we shifted to mineral sequestration research some of the money currently allocated to corn ethanol or underground injection of CO2 gas.