I'm assuming by food dyes you mean like the food coloring you can by in the baking asile of the grocery store.
I experimented last year w/ natural dyes from food. Onion skins boiled in water give a nice redish orange color, kinda brick like. Blue cabbage gave a beautiful robin's egg blue. Turmeric gave a nice golden yellow.
The egg in DD's hand was dyed w/ Turmeric
The one on the ground was dyed w/ lots of onion skins
that had been boiled for a while.
Strangely I don't have a pic of the red cabbage dyed one. It was my favorite.
You can google for more ideas.