Episode #125 of the Stack Overflow podcast is here. We talk Tilde Club and mechanical keyboards. Listen now
3 of 3 Shorter golf.

Python 3, 497 484 480 bytes

P=print
N=lambda m:"an"[:("E"==m[0])+1]+" "+m.lower()
h="I hear "
A=[a.split("|")for a in"Polar Bear|growl:Lion|roar:Hippopotamus|snort:Flamingo|flut:Zebra|bray:Boa Constrictor|hiss:Elephant|trumpet:Leopard|snarl:Peacock|yelp:Walrus|bellow:Zookeeper|whistl".split(":")]
for i in range(11):
 a,b=A[i]
 if i:P(h+N(a)+" "+b+"ing in my ear.\n")
 a+=", ";P(a+a+"what do you hear?")
P(h+"children...\n\n..."+",\n".join([z+"ing like "+N(y)for y,z in A[:-1]])+"...\nthat's what I hear.")

(Credit for 13 bytes saved should go to Gábor Fekete, and an additional 4 bytes were saved by Taylor Lopez.)

So you could probably guess I was going to use a language like Python as a teaching tool. This was the program before I started golfing it.

def startsWithVowel(name):
    firstLetter = name[0]
    return firstLetter in ["A", "E", "I", "O", "U"]

def aOrAn(name):
    return "an" if startsWithVowel(name) else "a"

animals = [
#    NAME (0),     SOUND (1)
    ("Polar Bear", "growling"),
    ("Lion", "roaring"),
    ("Hippopotamus", "snorting"),
    ("Flamingo", "fluting"),
    ("Zebra", "braying"),
    ("Boa Constrictor", "hissing"),
    ("Elephant", "trumpeting"),
    ("Leopard", "snarling"),
    ("Peacock", "yelping"),
    ("Walrus", "bellowing"),
    ("Robot Dragon", "screeching"),
    ("Owl", "hooting"),
    ("Monkey", "laughing"),    
    ("Zookeeper", "whistling")]

NAME = 0
SOUND = 1
for animalNumber in range(len(animals)):
    thisAnimal = animals[animalNumber]
    if animalNumber + 1 < len(animals):    
        nextAnimal = animals[animalNumber + 1] 
    else: 
        nextAnimal = None

    print("{n}, {n}, what do you hear?".format(n=thisAnimal[NAME]))

    if nextAnimal != None:
        print("I hear {a} {n} {s} in my ear.".format(n=nextAnimal[NAME].lower(),
                                                     s=nextAnimal[SOUND],
                                                     a=aOrAn(nextAnimal[NAME])))
    else:
        print("I hear children...")

        children = []        
        for animal in animals[:-1]:
            children.append("{s} like {a} {n}".format(n=animal[NAME].lower(), 
                                                      s=animal[SOUND],
                                                      a=aOrAn(animal[NAME])))
        print("...{}...".format(",\n".join(children)))
        print("that's what I hear.")