Sticking fabric to wood?

I would use polyurethane wood glue as it is waterproof for washboard channels could get wet. Sikaflex type polyurethane would also do the job and may be less messy than wood glue.
 
If its just lining the channels, why not get some sticky backed Velcro, cut to width and just use the softer loop side.
 
all purpose building adhesive obtainable from Wickes in tubes, waterproof, white, spreads easily as you do not want lumps & can be placed in corners easily with the thin nozzle
has reasonable grab characteristics but still allows one to slide the item into place & stretch the fabric properly. Sticks to teak whereas a lot of products will not
 
OK, the bottom line is that epoxy works best if applied with a non-porous roller, as used for wallpaper or printing ink.

Here is the result, first a mock-up, then the washboard mounts close-up.

Washboard-Rack2_zpslrbejfig.png~original
Washboard-Rack1_zpsud7fxqs9.png~original
 
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