Oh Ninja Turtles 3, I love you so much.
A lot of other games that could fall into the category the same way Bad Dudes does as a sort of beat em up, but I would say they're more of "action platformers." Good examples would be Castlevania, Werewolf: The Last Warrior, and Ninja Gaiden. Then there are the games that have a few levels that are beat em ups, then levels that aren't. Examples of that would be Bayou Billy or Battletoads.
Don't forget to check out part one!
Next up: The 16-Bit Consoles Super Nintendo & Sega Genesis!