I don't mind this idea, but I think the rules governing it should be quite severe, to limit it to situations where a team feels its genuinely been dudded in a circumstance that has the potential to affect the games outcomes. I'd suggest each team may appeal one decision when batting and one decision when bowling, per game, not innings and I think only captains should be able to appeal a decision - not the batsman or bowler in question. The criteria for overturning a decision has to be that the decision is clearly wrong - ie if the 3rd umpire is at all uncertain, they should defer to original decision.
Only being able to do so once will deter teams from using it unless they think its a clear mistake - you'd hate to waste it on the offchance of getting a decision overturned, and then have a hideous decision made against you and be powerless to do anything.
Captains would no doubt apply some strategic thinking about what to appeal or not, but that's doesn't make it a bad idea - losing your number 11 to a wrong call when you've got 450 on the board in the first innings isn't as big a deal as losing your star batsman at 4/90 chasing a fourth innings total - and this option should only be exercised in drastic circumstances.
I think it would significantly improve the relationship between umpires and players, provided one umpire didn't make a succession of decisions that were overturned on appeal - but then, that umpire would surely have to be in difficulties to retain their place on the panel. I believe it would improve relations because:
a) providing somebody with a method within the rules to properly ask for confirmation of a decision does not have to diminish the umpire's authority - the existing rules requiring respect towards umpires would remain in place, and its within the rules - hence you are not in fact seeking to get the umpire to do something improperly (this is a slightly circular point, but I think, if you follow it through, it does make sense), and
b) the option of the appeal should quell a lot of the dissent that currently occurs. Think about the dummy spits some umpires endure. If the play really feels wronged, they, through their captain, should appeal. If they don't think its worth using the appeal, get on with it. It gives the umpire a neat answer to challenges they get - "You think I'm wrong, go on, appeal it!".
If doubts exist, as they appear to, regarding hawkeye, then don't let the third umpire use it - slow mo replays, particularly with the highlighted 'strike zone' show all you really need anyway.
Giving the third umpire the responsibility to call no-balls, and the umpires earpieces for stump mike are two sensible suggestions that would assist the officiating umpires to get it right.
The old saying about things working out in the long run is a good one, and I don't think this is a change the ICC NEED to make, but I think it is one they CAN make that should improve the game. They can bring it in as a trial and if it doesn't, like the supersubs, it can always be abandoned at the end of the trial period. Changing the rules of the game does require care, but the rules have never been set in stone, and lots of the changes over the years have improved the game.