How do you clean your cast iron cookware?

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I love cooking meat, chicken, eggplant, etc. on my Lodge Pro-Grid Cast Iron Grill/Griddle Reversible Pan (picture included). What I hate doing is cleaning it afterwards, especially the grill side of it and when there is burnt and charred stuff on it. If anyone uses this type of cookware, how do you guys clean it? And how do you remove all that char on it, while keeping it from scratching and without damaging the seasoning? Last night, my husband made chicken breast that was soaked in teriyaki type sauce on the grill side of the pan. He even poured the remaining sauce from the bowl onto the pan. After we removed the chicken breast there was a lot of stuck on and charred black stuff on it.... He cleaned it yesterday but when I took it out today I saw that a lot of the black stuff never came off. I tried using a sponge and Palmolive dish washing liquid. And I even used a chopstick to try and remove all the gunk. Plenty came off but the rest is not coming off. And I've been at it for a while...