With the furring strips essentially 12" oc, when you hang the rock you can either screw into every other strip for 24"oc or into each one for 12" oc.
The former is a less-desired option, and I'd do it only if usiing 5/8ths inch rock. The latter is a heck of a lot of screws to drive and screw heads to mud.
If you don't want to tear down the furring strips, I'd run 12" oc and you could use half-inch board. Before rocking, I'd check the security of the furring strip-to-joist connection, you may want to pop a couple of ring shank nails through the furriing strips and into the joists for added holding power. With acoustical ceilings, some installers simply staple the furring strips to the joists instead of nailing or screwing. Check to insure you have a secure connection before you rock.
Check the integrity of the furring strips as well. Sometimes the wood may be a little brittle and when you hang the rock, the drywall screws can simply split the strips. If they are suspect, you may want to tear them down, install a sound attenuating channel (optional) perpendicular to the joists, 16" oc, and screw the rock to the channel. The channel would help prevent sound transmission between the uppstairs and the downstairs.