I was able to figure this out and wanted to post it in case any other Rackspace cloud users stumble in.
First, the download link contains two DLLs. Be sure to use the "FlickrNet.dll" for your web application.
Also, you will probably have to change the Flickr Cache location. You can do this within the Web.config by using the following code
This should point to a directory you have permission to write to. This also appears to be an absolute path (C:\directory), so if you can't figure out that path within a cluster or cloud, simply build a test page with the following:
Response.Write( Server.MapPath("~/directory/") );
Just put that in place of your "path" in the web.config and you should be ready to go!