How long cockroaches live varies between species, which can average between 5 weeks to 3 or more years.
Researching university etymology departments, here's a collection of data for the life spans of a few cockroaches common to the United States.
According to the entomology department at Penn State University, the German cockroach (Blatella germanica) has an average lifespan of 20-30 weeks.
"German Cockroaches - Entomology - Penn State University." Entomology - Penn State University. N.p., n.d. Web. 4 Jan. 2012. .
According to the University of Rhode Island's Landscape Horticultural Progarm, the brownbanded cockroach (Supella longipalpa) has an average lifespan of 10 months.
"Cockroaches." The University of Rhode Island. N.p., n.d. Web. 4 Jan. 2012.
The entomology and nematology department at the University of Florida identify the average lifespan of the American cockroach (Periplaneta American) to be 2-3 years.
American Cockroach Periplaneta americana 2-3 years
"American Cockroach." Department of Entomology and Nematology. University of Florida, n.d. Web. 4 Jan. 2012.
Blatta orientalis L.
According to Penn State's Entomology department the Oriental cockroach (Blatta orientalis L.) will live an average of 5-26 weeks.
"University of Kentucky Entomology Youth Resources - Pet Bugs." University of Kentucky. N.p., n.d. Web. 4 Jan. 2012. <http://www.uky.edu/Ag/CritterFiles/casefile/ /bugconnection/petbugs/exotic/exotic.htm#roaches>