Cockroaches ARE dirty.
If by definition you consider disease, bacteria, and virus transmission dirty, then yes, cockroaches are filthy.
Because cockroaches travel indifferently between waste and food, they pick up organisms on their legs and bodies spreading them wherever they populate. Cockroaches are linked with the spread of food poisoning such as Salmonella spp. and Shigella spp. They may also spread the hepatitis virus, coliform bacteria, Staphylococcus spp., and Streptococcus spp They have also been linked to the spread of typhoid and dysentery.
“Medical Significance of Cockroaches: Disease Transmission .” Department of Entymology. North Carolina State University, n.d. Web. 15 Dec. 2011. <www.cals.ncsu.edu/entomology/schal_lab/DiseaseTransmission>.
“Cockroaches.” University of California Agriculture and Natural Resources. N.p., n.d. Web. 15 Dec. 2011. <www.ipm.ucdavis.edu/PMG/PESTNOTES/pn7467.html>Top of Form