Guess why? Chicken thigh.
Guess when? Chicken hen.
Guess whether? Chicken feather.
Guess who? Chicken poo.
*Need more examples fast. *
Guess how? Chicken kung pao
Guess what for? Chicken and cole slaw.
Okay, I’ll get my coat.
Guess not, Chicken Snot.
Just out of curiosity, Lemur866, will you be hosting a three-year-old?
Guess what gloop? Chicken soup.
Guess which meat? Chicken feet!
Guess what matches? Chicken scratches.
Guess what’s got poop? A chicken coop!
Guess who gets a rest? Chicken in nest.
Guess which mascot? Chicken with ascot.
Guess first male? Chicken entail. [okay, that one’s geared towards adults.]
Guess which bling? Chicken a la King.
Guess which day? Chicken lay.
Guess which leg? Chicken egg.
Guess what curves like a crest? Chicken breast.
Guess a racetrack winner… Chicken TV Dinner.
Guess I’ll argue who? Chicken barbecue.
Guess genders… chicken tenders.
Bravo!
Guess where to sit? In chicken shit.
Guess who’ll take ya? Chicken cloaca!
Oops - I killed your thread.
Guess nothing more can be said.
(Chicken’s dead.)
Guess what?
Chicken Butt.
Know why?
Got Dookie in your eye.
You people are all insane.
[delChicken brain![/del]
Now that was freaking hilarious.
(chicken aquarius?)
Guess so, Chicken dough.
So, is this something that very young kids find funny these days?
It would be nice of the OPer, Lemur866, to clarify.
What bugs me is how they all rhyme except “what” and “butt”. No way in the world those words can rhyme. 
If my kids three year obsession with “guess what?” “chicken butt” is any indication - its high humor in the 5 to 10 year old set.
(It especially causes them to howl when they say “guess what, mom? guess what!?” when they are trying to tell me something about there day and I say “oh, I don’t know…CHICKEN BUTT!!!” Yeah, I’m that kind of mom.)
Over by me, what is pronounced wut and butt is pronounced but. Viola, a rhyme!
Guess the plan? Coq au vin!
What’s this thing? Chicken a la king.
What’ll it take? Chicken fried steak.
What’s up, yo? Arroz con pollo.
Guess how they test the durability of a jet engine under extreme avian impact events, son?
Chicken gun!!
Are scalar chickens as real as they say?
Oh, they’re 1920s-style chicken rays.
Oh, they’re 1920s-style chicken rays.
Oh, they’re 1920s-style chicken rays.