I think the best choice would be 2x3 strapping, here is my rationale.
It would be very expensive to run ceiling joists between the beams, parallel to the beams, so let's rule that out off the bat. You will want to run something perpendicular to the beams.
You could install blocking between the beams - bottom edge of the blocking even with the bottom edge of the beams - or strapping under the beams.
The 2x6 blocking considered in the question is overkill. Since this blocking only needs to support the weight of the drywall, you could go with something much smaller, and you'd only need to toe nail it, joist hangers not necessary. I think you'd want 2x3 blocking spaced 16" OC, and you'd probably put the 3" face down, so you have more surface to screw into at the joints.
Strapping is usually done with 1x3's spaced 16" OC, but that's normally across 16" or 24" joist spans. I think you'd want to bump up to 2x3 strapping to avoid sagging. There is an added bonus with thicker strapping: you get a decent amount of space between the drywall and the beams. Properly protected, you can run pipes and wires perpendicular to the beams in this space - a big advantage when finishing a basement.
So with strapping instead of blocking, you
- lose a little ceiling height
- gain some wiring / plumbing space
- save a lot of time installing
To me, strapping wins.