Do cockroaches constantly fart?

Not constantly. About every 15 minutes.

http://wiki.answers.com/Q/How_often_do_Cockroaches_fart

(star_gazer, where you come up with these questions is beyond me)

How about a link that has some credibility?

Not that these links are all that credible:

Question: Do insects pass flatulence? It's a rather twisted question, but the thought has often entered my mind.

Answer: Um . . . this might depend on how you define "flatulence." An insect's digestive system bears little resemblance to our own in its structural details, even if the basic process is the same. Flatulence in humans is largely due to the activity of bacteria in our intestines, releasing methane gas. But there are very few insects whose guts contain methane-producing bacteria. In a practical sense, therefore, the only insects for whom the answer is "yes" would be termites, some wood roaches, and a few wood-boring moth and beetle larvae.

Those termites are something else, though. For more on termite and other animal flatulence, see:

http://www.straightdope.com/columns/read/1910/do-insects-pass-gas

And then there's this:

Insect flatulence may account for one-fifth of all the methane emissions on this planet. (Termites are also prodigious farters; indeed, dogs trained to sniff out termites are actually following their farts.)

Cockroaches are among the biggest contributors to global warming, since they break wind every fifteen minutes. Furthermore, they continue to release methane gas for eighteen hours after they die.

https://sabrinaschool.wordpress.com/tag/cockroach/