Hi sneather, on Petr's script collection page that bemfarmer mentions above, there is a script that will convert a face selection into an edge selection of all its surrounding edges.

It's this one here:
http://kyticka.webzdarma.cz/3d/moi/#SwitchSelectionFromFaceToEdges

Set up that script on a shortcut key (paste the whole script in as the "command" part of the shortcut key) and then when you trigger that key you should get the selection behavior that you are asking about.

But one note - you mentioned doing this so that you could add fillets as one example, but fillets actually understand a face selection already - if you have faces selected and then do a fillet all the edges belonging to those faces will be targeted for filleting. So if you want to do filleting there is no need to do this selection switching as another extra step, just run fillet directly after you have done your face selection.

- Michael