You could run 2x6's under the ceiling side to side by installing a ledger on either side and using joist hangers. This will be far less work than trying to attach to the ceiling, and you don't have to worry about compromising the ceiling, fasteners rated for overhead use, etc. You can use the same types of fasteners used to attach ledgers for decks to concrete walls. That will be more than adequate.
Just make sure you have a plan for moisture to escape, and use pressure treated lumber where ever it's in contact with the concrete, this might be the trickiest part of the job. Strapping under the ceiling joists might be a big help.
One thing to watch for. Pressure treated lumber is sold very wet and green and shrinks quite a bit. It would be tempting to use non-treated lumber for the ceiling joists, but I see wood in that space getting pretty damp even without direct contact with the concrete. This might be another advantage of the strapping, it will allow for some movement of the wood without messing up the drywall.
It all might be enough to make you want to install a drop ceiling - nobody installs them because they look nice, but they do make a lot of things simpler.