I would like to see it go beyond a visual animation

Monk (dps): does x damage, increasing the damage the mob takes by y for the next z seconds
dragoon (burst): huge mega jump that does pretty much the damage of the LBs now
white mage (heals): same as healing LBs now, revive on 3
scholar (heal + shield): heals 25-40% as much as white mages, provides a bubble for 30-50%, revive on 3
warrior (soak tank): Increases the parties hp by xx%, damaging abilities heal the caster xx%
paladin (defensive tank): lowers the damage the party takes by xx% (probably a bubble as well)
summoner (DoT class): Meteor storm type, doing damage to a large area over time (more damage total than blackmage)
black mage (burst AoE): large meteor doing large amount of burst damage to the target area

I'd also be all for each class having its own limit break bar even independent to them, however if, say, 2 DPS did their LB at the same time it would combine into an effect more powerful than the sum of its parts, or a tank and healer and so on.