The predominant finish on kitchenware is nothing at all. This is because bare wood inhibits bacterial growth and is, therefore, naturally
hygienic. Kitchenware is also washed repeatedly and all finishes eventually wear away.
Use warm soapy water to keep your kitchenware clean. Scrubbing is fine and tends to smooth the wood over time. Avoid soaking breadboards for long
periods because they can distort a little (this is only temporary). Avoid putting wooden items in a dishwasher because they can get very hot.
Also, the drying cycle can bake the soaking wet wood dry which leads to cracking.
I do, however, put a finish on my pepper mills, salt mills and wooden bowls. I use cold-pressed raw linseed oil or, sometimes, shellac. A wipe with a
damp cloth should be all you need to do to keep them clean.
See also Robin Wood's excellent website for more details about the superiority of wood over plastic.