But after typing the importprivkey command, pasting the key and hitting Enter I don't get an error - nothing happens except after about 10 minutes an empty line shows up with a right-facing arrow. No balance shows up in the Overview tab, nor are there any transactions shown. However, there's a new address showing up in the address book, with the same timestamp as contained in my private key.
Am I close to getting it right?
Yes.
It sounds like the private keys that you are importing don't have any bitcoins associated with them at the moment.