I know that sindrome. A friend of mine keeps filling his Desktop up until there's no space there. It gets damn ugly!
As for your request.. I have a few comments about it.
Do you really want the Desktop to be cleaned exactly when you download the stuff? I mean, it'd make it very easy to save stuff, but way harder to get to it (since you'd have to browse through your computer to get to that other folder where the stuff would be stored).
Also, the other folder would get all the stuff you currently have on the desktop, which wouldn't help much with organization.
So, i have 3 suggestions:
- Have soft links for folders on your computer linked to the desktop.
- Have a program that would store stuff after it's been on the desktop for some time, and store it according to some pattern.
- Use a download manager like Flashget
Now... For option 1, you'd have the folders on your desktop, and when you dropped stuff into them, the stuff would be stored inside those folders, somewhere else in your computer. This'd make you have to organize your stuff, but it'd be as easy as drag and drop. Also, the folders would be stored somewhere else, but quite accessible through the desktop.
For option 2, you'd have something not as organized (since it's impossible for the program that examines the files to identify which ones are which by any other method than their files unless you add some kind of tags to the filenames), but easier to keep.
For option 3, the download manager would ask you which category that download was, and automaticly store it in the right place. (i think the download manager can also store stuff based on the site where the download comes from). This'd make drag-and-drop unnusable, and might not be exactly what you want, but it'd be the more organized solution.
Also, you could do like i do: instead of saving everything, just clicking "open" on the download box, and the stuff gets stored in the browser's temp directory

Now, which one do you prefer?