Prior to 6.1, getting a 3-man team with 6 racial bonuses was pretty much like winning the lottery. Post-6.1, I think there's a fairly straightforward process to build 3-man 6-racial-bonus teams, so thought I'd share it.
The process:
1. Activate one of each of the "odd" racial types: beast (Goldmane, Leorajh, Meatball), robot (Pleasure-Bot 8000, Ziri'ak), bird (Dawnseeker Rukaryx, Talon Guard Kurekk, Talonpriest Ishaal), or ogre (Dagg, Tormmok, Blook). Ideally, you do NOT want to use a follower with a trait that won't get rerolled (such as Bodyguard). These are your "anchors" you will build your teams around.
2. Do missions for racial bonus items for the above races (e.g. You, Robot, etc.) and hang on to any items you get.
3. Do Follower Trait rerolls on your "anchors" until you get a follower with 2 racial bonuses (any two will work). Prioritize the type you have the most missions finished from step 2 (i.e. if you have 2 Bird Watchers ready to go, prioritize rerolling your bird's traits). Keep spending your rerolls on this until each of the four types has two racial bonuses.
4. Using the tavern (or rerolls), recruit the racial pair of the odd type's bonuses. Example: you reroll Talon Guard Kurekk until he likes orcs and undead. Use the tavern (or rerolls) to get an orc that likes undead and an undead that likes orcs. Getting single racial bonuses isn't hard at all.
5. Apply the mission items (bird watcher, grease monkey guide, ogre buddy, etc.) to the followers from step 4. Two in three times, you'll get a double racial bonus. One in three times, you'll fail and have to keep doing step 2 and step 4.
6. Reroll counters using Follower Ability rerolls until you get 6 nonoverlapping counters on each team.
The end result should be four 3-man 6-racial-bonus teams that each uses one of the odd racial types as an anchor. Although this seems complicated, it's really just a matter of time and way easier than trying to randomly reroll into a 6-racial-bonus team.
An example of the process:
* Activate Talon Guard Kurekk.
* Reroll until he gets two racial bonuses (i.e. orcs and undead).
* Get an orc who likes undead and an undead who likes orcs.
* Do missions for the Bird Watcher item.
* Add the bird watcher trait to the orc and the undead and hope it doesn't overwrite their racials.
* Reroll the counters to get 6 non-overlapping counters you're happy with.