I'm told "Twist and Shout" works well as a chantey, and I've also heard "The Itsy Bitsy Spider" used.
There are still people writing new chanties. I have just heard a tape by David LoVine (thanks, chanteyranger!) that consists entirely of songs and chanties that he wrote about the Lady Washington. The chanties sound traditional but are not, the words pertain to the ship and her crew, and they sound perfectly useful as work songs. Great tape, by the way.
I think traditional chanties sound the way they do because they tended to reflect the popular music of the time. Cranky Yankee's story about the teenagers is the modern-day version of the same thing.
Not to mention that traditional chanties get new verses all the time.