louglas,
Each Test Case has a collection of Results associated to it.
If you are looking for something from within Agile Central, you can use a Custom List set to Test Case Result and enter a query that limits the Creation Date to the range of the Iteration. There is an export option available on that page.
If you are looking for a way to do it via the API, you would need to query for the Test Cases and for each Test Case query for the collection of Results and then save the ones you are interested in using.
There may be helpful examples on GitHub or Stackoverflow if you search there.
Michael