Quick Draw (the feat) does not allow you to sheath or stow a weapon or shield any faster than normal. If you can't find a way to hold three shields or to store one as a free action (Glove of Storing, if your shield is less than 20 pounds), I don't think you can do this.
A Quickdraw Shield can only be stowed as a swift action when combined with a regular move, but if you have the Quickdraw feat, you can stow it as a free action, so that might be the way to make this work.
A DM might limit how many shields you can have stowed on your back at any one time, so you might have to have two returning shields and one quickdraw returning shield. You'd have one shield in each hand and one on your back.
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Though if they're Light Shields, couldn't he strap one to his wrist when it returns, then catch the other two in his hands?
You may want to consider the Blinkback Belt, though, as it's much easier on weapon choices and mobile.
Readying (strapping on) a shield is only a move action.
Quickdraw shields let you do this as a free action if you have the Quickdraw feat. Since Quickdraw Shields are light shields, and a light shield allows you to hold (not wield) another item in your shield hand, that would work too. You'd start with one hand wearing a quickdraw shield and holding another, and your primary hand holding a third. Throw the primary hand shield, take the held shield from the other hand, throw it, remove the quickdraw shield as a free action, throw that.
Next turn, catch the quickdraw shield as a free action, don it as a free action, catch a second shield as a free action, transfer it to the hand wearing the quickdraw shield as a free action, catch the third shield. Repeat.
Profit.