Nail firring strips to the bottom of joists that extend slightly below the
cross bracing and attach your sheet rock. This saves the cross pieces, but
puts somes "holes" in the joists from the strips - not a bad solution.
Alternate: remove the cross bracing - cut to the proper size and put back in
place. The original builder would have done so if they knew a ceiling was
going up.
Alternate 2: Worried about "originalality"? Don't put up the sheetrock.
Candy