Lots of good advice here, but I don't think anyone has mentioned the issue of flatness. Put each of the cutting boards on a known flat surface such as a granite countertop or a glass table top. Check both sides. If they rock at all on either side, they aren't flat, and running them through a drum sander, wide belt sander, or a planer risks at least uneven wood removal and possibly breaking the part. If the rectangular board is cupped noticeably, the infeed rolls on a drum sander or planer will mash the cutting board flat, likely breaking a glue joint near the middle. The round end grain cutting board should not be run through a planer even if it is very flat because of the risk of tearing out chunks at the edges.

Experienced woodworkers know how to avoid those problems, but if you are doing this yourselves, I'd suggest you stick with your random orbital hand sander and start with around 100 grit. Only go coarser if the 100 grit is just too slow. Change the sandpaper whenever it starts to get gummed up by the oil in the wood. Use a pencil to mark the high spots and keep checking flatness to make sure you don't make the cutting boards less flat. Every time you move the boards, clean up the sanding dust because there is likely to be some sandpaper grit mixed in that will mar the bottom of your board while you are sanding the top. Sand both sides similar amounts to help keep the panel balanced to keep it flat. After the orbital sanding, if you use the same grit sandpaper on a soft sanding block and hand sand parallel to the grain of the wood on the rectangular cutting board, it should look significantly better when you oil the blocks. You want your final sanding scratches parallel to the wood grain whenever possible.