I used food coloring to color some kids blocks that I made, but that was about 60 years ago, so neither the pictures nor the blocks are still around. Food coloring is readily available, and likely already in your kitchen closet, so easy to try on some of your wood. Not all wood takes the color the same, so a little experimenting and time of soaking might be needed, but food coloring is about as safe as you can get.
aniline is what I use, for food grade food color dye does work just less control to the tint,, look on ETSY there are ppl that sell organic/vegetable based dyes there.
Food coloring dye will eventually fade. Anyline dyes are very carcinogenic and shouldn't be used near food. I would consider using contrasting wood species, such as Walnut, Maple, Cherry, and Holly.
My wife uses RIT dyes exclusively for dying reed and other woods for basket weaving. What she worked might be more porous than what your working with, so best to try on a small piece first. You might need to let it soak for a while in hot dye-water.
I have used powdered RIT dye dissolved in DNA to dye wooden ornaments but are mostly baltic birch plywood. Woods typically used in cutting boards are dense so I doubt they would absorb much or any dye.
I have a large-ish vacuum chamber (about 18" diameter) that I use for cactus-juice wood hardener and for degassing epoxy. It helps get the liquid hardener fairly deep into the wood. This could also work.