You can compost that sod, whether the grass will regrow when there is sufficient moisture will depend on which grass it is. Most to the desireable grasses as so wimply that they die off easily while the invasive grasses do not.
However, there was no need at all to remove that sod to create new planting beds and by removing the sod you removed most of the best soil for those planting beds and most of the nutrients as well as the Soil Food Web in the area of those beds. all that is necessary to do to develop new planting beds is to cover the area with newspaper, cover the newspaper with a mulch, and let the grass (and "weeds") die and return the nutrients they took out of your soil to the soil when they do die.

If the stuff is truly dead you are o.k. (except for whatever seeds are in there). Some plants that look quite dead are not however. Your concern is probably about the stuff growing back.

A couple of things you can do.
1. Bang the sod pieces to knock off loose soil into a wheel barrow or whatever.
2. Stack the sod pieces upsidedown to break down and have a little bit less chance of new roots growing.
3. Stack the sod pieces upsidedown but put a few sheets of newspaper or cardboard between each piece of sod.

Your other concern is any seeds that are in there. After you have thoroughly killed the actual grass plants you can hot compost to kill the seeds or let them germinate a bit and then repeat the steps above. But that's a lot of extra work just to get rid of some sod. I would probably just kill the existing grass in the sod pieces and then add them to your compost pile or whatever.