American cockroaches (Periplaneta americana) are omnivorous and opportunistic feeders, and as a result, they eat a large variety of foods. They will eat almost anything.
Favorite foods: They are particularly fond of fermenting foods$^{4,6}$ and also prefer sweets$^2$.
Common foods they eat: plant materials, dead animals, sugar, oil, cheese, beer, tea, soap, leather, bakery products, paper, starch in book bindings, manuscripts, glue, hair, flakes of dried skin, soiled clothing, glossy paper with starch sizing and generally most human or pet moist food or other organic matter $^{1,3,5}$. They have also been observed feeding on dead or wounded cockroaches of their own or other species as well as their own cast-off skins and egg-capsules$^6$.
According to EOL:
Some of [their food] items, such as cellulose, can not be digested by normal means. However, like cows and other grazing animals, cockroaches have a symbiotic relationship with a bacteria that allows them to digest such substances.
- As a result, they could almost certainly survive on just wood and water.
Citations:
{1} Bell, William J. and K.G. Adiyodi. (1981). American Cockroach. Springer. pp. 1, 4. ISBN 978-0-412-16140-7.
{2} Barbara, K. A. 2014. American cockroach. University of Florida Institute of Food and Agricultural Sciences. Retrieved 11 February 2017. http://entnemdept.ufl.edu/creatures/urban/roaches/American_cockroach.htm .
{3} Jacobs, Steve. "American Cockroaches". The Pennsylvania State University. Retrieved 11 February 2017. http://ento.psu.edu/extension/factsheets/american-cockroaches .
{4} Jones, Susan C. 2008. Agricultural and Natural Resources Fact Sheet: American Cockroach (HYG-2096-08). Ohio State University. http://ohioline.osu.edu/hyg-fact/2000/pdf/2096.pdf.
{5} Milne, L. & M. Milne. 1980. National Audubon Society Field Guide to Insects and Spiders. Knopf, New York, NY. p 393. ISBN: 0-394-50763-0.
{6} Periplaneta americana: Palmetto bug. Encylopedia of Life. Retrieved 11 February 2017. http://eol.org/pages/1076920/hierarchy_entries/51102706/details .