Cockroaches DO bite.
Though cockroaches do bite, they are not an aggressive species and do not actively pursue humans as a food source.
Cockroach Bites Were Common on Ships
Centuries ago, sailors were forced to wear gloves and shoes so as not to be bitten by the cockroaches that frequently infested the old wooden sailing ships of the times. In an 1870 article published in the journal Nature, the author describes his personal experience with ship voyage infested with cockroaches. He reported that the sailors’ fingernails and skin appeared to have been attacked. Later that evening when he fell asleep he found that the cockroaches were making a meal of his skin as well devouring the entire hard skin of the heal down to the tender pink flesh. [Nicols, 1870]
Not a Large Threat Today
If a cockroach does bite, it is generally when the victim is immobile for very long periods such as during sleep and is rarely serious. Cockroaches will feed on anything edible and have been reported as feeding on fingernails, eyelashes, any calloused sections on the hands and feet, and food residue around the mouth.
“Cockroach Facts.” Welcome to marlin | marlin. University of Massachusetts, n.d. Web. 27 Dec. 2011. http://www.bio.umass.edu/biology/kunkel/cockroach_faq.html#Q7
“Urban Entymology Cockroaches.” UC Riverside Entymology. N.p., n.d. Web. 27 Dec. 2011. <entomology.ucr.edu/ebeling/ebeling6.html#cockroach bites>.
Nicols, Arthur. “The Cockroach.” Nature 3.58 (1870): 108. Print.