I've stayed at both resorts (several times at the Polynesian, once at AKL) and they're my top two favorite WDW resorts.
I agree that the transportation at AKL was a total non-issue for us. The buses were quick and plentiful. I thought the theme of the resort was outstanding and almost elegant in an African-theme sort of way, and I liked it even better than the Wilderness Lodge (And I definitely loved our stay at the WL too!). I would never try to dissuade anyone from staying at the Animal Kingdom Lodge, especially if they or their children wanted to see live animals around their resort every day!
Having said all that, I LOVE LOVE LOVE the Polynesian resort and in my own personal opinion, NOTHING comes even close.
With the ages of your children, I would recommend the Polynesian (again, UNLESS they have a strong desire to see animals every day, in which case I'd then recommend AKL). You'll find it so easy to choose between the monorail or the resort boat to get back and forth to the Magic Kingdom. There's a path to go right next door to the TTC (Ticket & Transportation Center) to right onto the Epcot Monorail. You will have to take a bus to get to Disney Studios and to AKL.
The rooms at the Polynesian, as already mentioned, are among the largest on Disney property (comparing standard rooms, not suites). With the ages of your children, I would think that bunk beds would not be something you'd be interested in because I'm not sure that it's safe for a 3 year old to be in a top bunk, so the issue of bunk bed availability wouldn't come into play. However, in every room at the Polynesian there are 2 queen beds PLUS a twin-sized day bed. Plenty of room for your family.
While I agree that Jiko is outstanding and that there is no Signature Dining Restaurant at the Polynesian, I'm not sure if that's something that would effect your decision because of the ages of your children. That's not to say that people with young children shouldn't or wouldn't go to a Signature Dining restaurant, but many don't. I know that 'Ohana would be a lot of fun for your kids, and the Kona Cafe is my familys' all-around favorite Disney restaurant. Also, 'Ohana offers a character breakfast each morning, if that's something you'd want to consider.
I've taken tons of pictures of the Polynesian resort (as well as AKL and every other resort we've stayed at). You can see them by clicking the link below in my signature, then clicking on "Disney World Trips".
I don't think you'll go wrong choosing either resort, but my final recommendation is for the Polynesian

Have a great trip, no matter where you decide on!!!!
