The only real way to do it is to screen it -- you use a series of trays and screens of varios sizes so that only soil smaller than a given size can pass through.
When I did a small patch at my home, (putting in a french drain), I used a bread tray that I had fit with ~1cm hardware mesh to reduce the size of the holes, but that took two of us to shake it, and a fair bit of work. If I was doing a large amount, I'd build something to suspend the frames, so you could load a bucket of soil in, shake it, then dump the screened debris to the side (as the soil will fall straight through).
The clay is going to bring other problems -- if you don't load the screens too much, and get the rocks bumping into each other, it will hopefully knock off the clay, and fall through; if you want to be really thorough, you'd wash the dirt through the screen (eg, if you were trying to do a soil analysis, which you're not), but it'd make a real mess, so I wouldn't recommend it.
Depending on the size of the rubble, I might go with two screens -- one about 3" / 7.5cm, to catch the really large items and the other between 1 and 2 cm (~0.5") to catch gravel sized bits.
From the sounds of things (sticky clay) this will not be fast work. I'd try to figure out the technique just digging by hand, and test everything to find out how fast things will go, and then consider getting some friends in to help shake the trays, move the finished stuff around, etc, so if you are renting a machine and/or hiring an operator for it, you don't have to go through the learning process while you're wasting time on the excavator.
You might have to deal with more than one layer -- if there's an obvious topsoil layer, and a clay layer, you'll want to handle them separately, so you're not putting too much clay on top for planting; for this it might be worth hiring someone experienced who can be more precise in the removal. (I'm not sure how difficult it is to operate the excavator, but if you're spending the first few hours trying to learn how to do things half-decent, it's potentially a lot of lost time if you're trying to get this done over a weekend) Hopefully, an experienced person will also do less harm to the roots of trees -- if nothing else, their experience might tell them how close they can get before you have to switch to hand tools.