Its pretty well known how plowpan reduces crop yields and is why many farmers subsoil their fields. Plowpan is the hard layer of soil 6-7 inches down caused by the repeated plowing at the same depth. Subsoiling breaks up that layer http://www.youtube.com/watch?v=8VKsK3u_JWE
Image taken from: http://www.soilandhealth.org/03sov/0302hsted/030201/03020100frame.html
Note that the units are in feet.
Roots can go deeper than you would think. Plants can do ok in pots, but do better when their roots are allowed plenty of space to grow.
See here and click on the crop of your choice: http://www.soilandhealth.org/01aglibrary/010137veg.roots/010137toc.html
It is fairly difficult to find a vegetable with roots only occupying the top 6 inches of soil. Most can easily go down 2ft.
But, as far as drainage goes... If you dug a hole in the yard as big and deep as those frames are and filled the hole with water, I bet the frames on the concrete would drain faster than the hole in the yard. So drainage on the concrete is probably not a big deal unless the frames make a fairly water-tight seal. If in doubt, stick a 1/4 inch shim under each corner of the boxes, or drill big holes. One good thing about the concrete, it will never become water-logged. It will drain the same no matter where your water table is and no matter how long it rains.
As a test, fill as many buckets as you can and then dump all the buckets in the frames at the same time and see how long it takes to drain.
Vermiculite is good stuff. Compost and moss will decay and disappear, so you will have to add it as the level drops. Alternatively, there could be someone in your area who sells a nice garden soil. The price here is $20-$35 a pickup truck load for some nice black soil. I'd check craigslist. As an illustration, here's an ad in my area http://chattanooga.craigslist.org/grd/3924301730.html
Vegetable Garden Soil - $26
We sell Blk Gld known as Windwood's Premium Planting Mix at Windwood Bulk Center on 5342 Battlefield Pkwy in Ringgold. Windwood's Premium Planting Mix is great for vegetables, flowers and shrubs. Just put down 6 sheets of white newspaper then 8 inches of Windwood's Premium Planting Mix and you have a flower garden right on top of your grass, no digging required. For a vegetable garden just put down 6 sheets of white newspaper then 12 inches of Windwood's Premium Planting Mix and plant your vegetables. Windwood's Premium Planting Mix is available with 6 months slow relaease fertilizer or organic with no fertilizer it is a weed free planting mix We can load your pick up or trailer or deliver.
There are many ways to go about this, it just depends what you want to accomplish. My only issue with the concrete is root depth. If you're ok with that, then go for it. The worst that will happen is you'll decide to drag the frames onto the grass later.