The short-cut method is to lay what you can W/O compressing the insulation being used. (If you do, you'll REDUCE the effective insulation value ...) And under the boards you don't have direct access to, you have a couple of ways to go.
1) You could 'slide' some bats UNDER the boards wherever you can. If you must, you can use < R-50 so as to prevent the insulation batts from getting 'stuck' on the boards as they're slid underneath. 2) You can 'cut out' every-so-many feet some of the boards so you can have access. In other words, you don't have to remove ALL the boards to do this job.
ANd if the insulation you use is blown, your Contractor will most likely cut out a few holes in the boards, or as I described, cut out small sections to gain access. After that, you can lay the 'pieces' of boards back down and screw them to the joists. Just set your saw-depth to the THICKNESS of the boards; no more. And before you screw the boards back down, make sure you drill pilot holes first.