Long ago, I remember reading with great curiosity Rabbi Kushner's Why Bad Things Happen to Good People? How disappointing that Kushner's intellectual answer seemed to be that God isn't omnipotent. (His practical answer not to worry about the question but just to do good is much better.) The idea of limiting divine attributes in part to answer the problem of evil has recently had some defense (e.g., here and in the work of open theists), so I guess it's time to blog the objection to Kushner—which applies to the others as well—that I had when I read him, with some elaboration.
Basically, the objection is that as long as God remains pretty good, pretty smart (he was smart enough to create us!) and powerful enough to communicate with us (Kushner at least accepts this), then serious cases of the Problem of Evil remain. Moreover, these cases do not seem significantly easier to solve than the cases of the Problem of Evil that were removed. Consequently, the intellectual benefit with regard to the Problem of Evil is small. And the intellectual loss with regard to the simplicity of the theory is great—the theory that God has all perfections is far simpler.
Start by considering a deity whose goodness is unlimited but whose knowledge and power are fairly limited.
Consider, first, the problem of polio. This is certainly a horrendous evil. And the limited deity could have alleviated a significant portion of the problem hundreds of years earlier simply by whispering into some people's ears how to make a vaccine—surely any deity smart enough to create this world would be smart enough to figure out how to make vaccines. Maybe the limited deity couldn't have prevented all cases, in the way that an unlimited God could. But given that neither did the wholesale prevention happen nor did the partial prevention by vaccines happen as early as it could have.
Consider, second, the many cases where innocent people suffered horrendously at the hands of attackers, where the attack could have been prevented if the people had been warned. Even a deity of limited power and knowledge should be able to see, for instance, that the Gestapo are talking about heading for such-and-such a house, and could then warn the occupants. (I am not saying that such warnings were never given—for all I know, they were in a number of cases. But I am saying that there are many cases where apparently they were not.)
Moreover, even if one limits the goodness of the deity, and only claims that he is pretty good, the problem remains. For unless the deity had a very serious reason not to tell people about vaccines and not to warn the innocent victims of horrendous attacks, it seems plausible that the deity did something quite bad in refraining from helping, so bad as to be incompatible with being pretty good. (If the deity had a reason that fell a little short of justifying the refraining, then that might be compatible with being pretty good; but a reason would have to be pretty serious for it to fall only a little short of justifying the refraining when the evils are so horrendous.) So even if one thinks that the deity has limited power and knowledge and is only pretty good, the problem of finding very serious reasons for the deity's non-interference remains.
Granted, the problem is diminished, especially if one has decreased the belief in divine goodness. But notice that the decrease in belief in divine goodness is the most religiously troubling aspect of a limited God doctrine. And even that does not make the problem go away.
Moreover, the sorts of things one can then plausibly say about the remaining problems of evil are things that, I suspect, the traditional theist can say as well about this and many other cases. Perhaps God does not prevent all attacks on innocent people (for all we know, he prevents many) because he wants humans to have effective freedom of will. Perhaps he wants to give victims opportunities for forgiveness of their aggressors in an afterlife. Perhaps God does not prevent disease because he wants us to help our neighbor and to develop medical science to this purpose. Or to give us an opportunity to join him on the cross in redeeming humankind. Or perhaps God prevents many evils, but his purposes do not allow him to prevent all, and some arbitrary line-drawing is needed. I am not saying that these answers are sufficient (though I think some contain a kernel of something right), but only that they can be equally used in the case of a limited and unlimited God, and in the case of an unlimited God such answers may well have rather general applicability.