Knowing that some alternate-universe version of me will kick a puppy doesn't make it all right for me to kick puppies, any more than knowing that George Bush is down with waterboarding makes it all right to torture prisoners. The right thing to do is the right thing to do, regardless of what anybody else does or doesn't do, in this universe or any other.
This argument suffers from the twin flaws of (i) resting on a blatantly false general principle, and (ii) badly misunderstanding the opposing arguments. I'll tackle the latter point first.
Nobody's arguing that it's okay to do bad things just because someone else does. It's bad to torture people, even if GWB does, for the obvious reason that mimicking his harmful actions will cause even more harm to result. But in the multiverse case, we are supposing that every possibility will be realized exactly once. This leads to a vital point of disanalogy. We are supposing that my otherworldly counterpart will kick a puppy if and only if I don't. So whether I kick the puppy or not makes no difference to the total outcome. Whatever I do, the multiverse as a whole will contain exactly the same number of kicked puppies. That is why, as a concerned advocate for the welfare of puppies everywhere (and not just in my corner of the multiverse), the multiverse hypothesis implies I have no reason to refrain from kicking puppies. It's a consequentialist argument, not a "But if Georgie did it why can't I?" whine.
As for the general principle: others' actions are in fact highly relevant to determining what one ought to do. This is because the ultimate outcome of my actions may depend in part on others. For example, if my neighbour is suffering from a rat infestation, I might well offer him some rat poison. But this is no longer so advisable if I learn that he desires to poison his wife. So it's just false to claim that right and wrong are determined "regardless of what anybody else does or doesn't do". Other actors can affect the downstream consequences of my actions, and downstream consequences matter.
Although Chad's stated argument fails, perhaps the underlying idea can be saved. In a non-consequentialist spirit, we might claim that one ought to be virtuous, even if this does no good. For example, some have claimed that we shouldn't invest in unethical industries (tobacco, etc.) even if our abstention drives the price down, causing others to invest correspondingly more in our place. To abstain in this way is to be what Vera Bradova calls a 'Moral Dupe': "one who makes sacrifices on behalf of his/her own conscience... and in so doing aids and abets those who do not."
My views on this haven't significantly changed since writing my earlier post on this topic, so I'll just echo the concluding passage:
The most plausible ethical views allow that actions have other morally relevant features besides consequences, but nevertheless recognize that these other features are ultimately grounded in consequential concerns. Ethics is important because (we assume) our choices can affect others and change the world. If this assumption is false, as modal realism [or the multiverse hypothesis] would have it, then our decisions - and hence the norms governing them, i.e. ethics - are inconsequential, in the most derogatory sense. To care about ethics even when it makes no difference would be arbitrary and fetishistic.
The only escape, I think, is to embrace a more 'partialist' value theory, and say that what matters are the consequences around here, never mind the puppies over yonder. I don't think much of that option, though.