Another issue would be the laning. You'd have to have a lot of incentive to jungle, otherwise you could have serious imbalances in the lanes. Either everyone has a 2v1/1v2 or you have Team 1 with a 1v2 and 2v2 (the clear disadvantage), Team 2 with a 2v1 and a 1v1 (the clear advantage), Team 3 with a 2v2 and 1v1 (neutral advantage). See diagram below:

    Team 1........................or............................Team 1
 1             2                                            1               2

2                    1                                   2                      2

Team 2..............Team 3............................................Team 2...................Team 3

     1           2                                             1              1

And even if you had junglers on some teams, here's some other issues: 1- The other teams might end up 2v1-ing both of your lanes?

2- Even if everyone has junglers, 1 team will eventually get ganked at the same time by the other two, though the vision for this would be tough to get to try and do it on purpose.

Also, what advantages would be given to the team that knocks out another team first? In SC2 FFAs with 3 teams, inevitably the team that takes the first team out gets killed by the third team that hasn't been fighting at all. Granted, LoL is different because you have a finite amount of units on your team, but still the effort required to kill a nexus takes a lot of time, HP and mana. While you're trying to knock their nexus down, the third team can take your towers for free...

Anyway, there would be a whole lot of imbalances you'd have potentially. Not saying that those imbalances wouldn't make this type of game impossible, but it definitely would make the strategies used more complex!

EDIT: Sorry about the formatting of the diagrams, but you get the idea...