Should I play the predecessors before Assassin's Creed III?

There are plenty of things you might not get if you played it without having an idea of the prequels. However, I wouldn't necessarily recommend playing them to get the story.

The first one is incredibly repetitive and involves many menial tasks whilst the second one struggles in aspects of the story line.

Strictly speaking, there won't necessarily be things you won't get, just things you won't notice. Most games are developed so that you would be able to understand the plot without having to play previous games (although I think most people would struggle with Metal Gear Solid 4).

I'd recommend either reading the wikis for the plots or there is a handy video here that will sum up everything brilliantly. (Obviously, don't watch if you wish to play the games instead)

It's not really necessary to play the previous games but if you do you'll surely have a deeper experience with the story. You can also take a look at The Assassin's Creed Wiki. The main games previous to the third are (in order): AC, AC2, AC Brotherhood and AC Revelations.

There is a very interesting article that answers this question: http://www.usgamer.net/articles/which-assassins-creed-should-i-start-with-a-gateway-guide

Basically, you can skip one and go for ACII. I started AC the first 8 years ago, but never finished it. I decided now to re-start and it looks like ACII is much better than ACI with faster action. Starting there won't affect you.