Luan subflooring was used on walls in kitchen and dining room in place of sheetrock in late 50s house. I'm replacing with sheetrock in dining room. Kitchen was remodeled over Luan (don't want to take it apart) but how do I handle backsplash and what little wall space is in kitchen? Luan is flimsy enough that it moves when you press on it so I don't think it will support backsplash covered in same laminate as countertop. Suggestions? Thanks.

You could take it off just where it's exposed. Use a sawzall or multi-tool for places like under upper cabinets (Milwaukee makes a mini-sawzall they call a Hackzall that is really good for delicate cuts in all sorts of places).
And you might want to add insulation down behind the plumbing or the whole exterior wall, if they didn't.
For resale value, if you are interested, use small glass tiles instead of laminate backsplash.

You can get a laminate back splash to match your counter glued to 3/4'' compressed board. This can be glued over existing surface.

Chuck