This is something that I learned as well. I also tend to wait a few seconds before accelerating from a red light because I fully assume that someone is going to be a complete asshole and try to blow the previously yellow light at a high rate of speed.
yes - I give a right-left check look when coming off a red in the first spot to ensure there's not some cross traffic who decided the rules don't apply.
A staffer at a local moto shop was near killed a few years back. He was the first vehicle into the intersection from his lanes (multi lane road) on a green while on the way to work. A stolen SUV on the run from the cops blew the red on the cross road and hit him full broad side at a high rate.
And related - never hurts to check right if you're turning right in an area with pedestrians - be you on bike or in car. I see tons of folks who don't think to look right when turning right on red/stop and near hit a ped. Or, being lucky me, hit me...