I accept that animals have rights, but I'm inclined to think that moral agency is a better candidate for rights-making than capacity for experience, because there clearly are rights that are due to someone because of their moral agency -- e.g., just desert -- and it's much less clear that anyone has rights due to capacity for experience. The reason for this, I think, is that rights are objects of justice: to secure a right is to follow an obligation of rendering something its due. There are dues that follow from moral agency; but it's much harder to argue that there are dues following from capacity to experience.
Animal rights, if they are to be grounded in anything, have to be grounded in something very like what human natural rights are grounded, because they can only be natural rights. Since non-rational animals are not moral agents, we can't be indebted to them out of what they morally deserve. But basic human rights are not desert-based rights, either. There do appear to be basic human rights based on moral agency that are not due to desert but due to common good: insofar as we are joined with others in a moral community through our rationality, we owe fellow members of that community certain things because the common good of a moral community establishes a certain sort of justice between its members.
So the moral agency theory is not wrong in the sense that it fails to identify a basis for rights; at most it's wrong in the sense that it is incomplete. And this, I think, is quite right; it is incomplete.
Rights are objects of justice; to protect a right is to conform to an obligation to render what is due to the possessor of the right. So the natural question, whenever we are looking at the basis of a right, is this: What sort of justice is involved?
Consider a basic human right, natural and unalienable, like the right to life. This is not a desert-based right; nor does it appear to be a right that we have in virtue of some common good. Perhaps some people may deserve that their lives be protected, or the common good may require that their lives be protected, but, if so, this would be redundant; in following our right-to-life obligations we do not need to refer ourselves either to the deserts of the individuals or the common good of the community to which they belong, but owe them something regardless of what they have done or whether it conduces to the good of the community.
It is very easy to express this using a civil theology. This is, in effect, what the U.S. Declaration of Independence does:
...they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness...
The idea being, of course, that there is a sort of justice that obtains between men that has as its object 'endowments' of the Creator; that this justice is not in any way optional or dependent on their being part of a community; and that this justice authorizes them to establish guarantees that the obligations will be met, guarantees like governments and magistrates, which secure rights. In any Lockean view, however broad it may be, this justice is a derivative justice; there is a due to be rendered to men because there is a due to be rendered to God. (The precise nature of this derivation in Locke is a very heated issue in Locke scholarship; and, of course, a broadly Lockean view need not regard the derivation in the same light Locke himself does. But any broadly Lockean view, I suggest, will have some version of this derivation.) And there is no question that the account presented in the Declaration is a broadly Lockean view.
The advantages of a civil-theology view are its relative simplicity and clarity. But note that there is nothing in this civil theology view that rules out animal rights or supports the moral-agency-only view. If even some basic human rights (the ones that are pre-communal and pre-agency) are derivative rights that human beings have in virtue of God's having rights, we can't rule out derivative rights for non-human animals in virtue of God's having rights -- for God has rights whether we are thinking about humans or non-humans. God's having rights establishes a sort of justice between human beings; something is due to our fellow creatures precisely because they are fellow creatures and we owe God His due (which we render in part by rendering what is due to our fellow creatures). An illustration: if A and B are in debt to C, and C requires that A and B pay it back by helping each other in some way, their debt to A, plus the requirement for paying it, establishes a derivative debt to each other. And it is clear that we can only rule out animal rights if our due rendered to the Creator is such that there is no derivative 'due' to them established by our 'due' to our Creator. Indeed, we can't even rule out rights for inanimate objects if there can be some type of justice shown by practical reason to exist between us and them. If there is justice, there is an object of justice; and an object of justice is a right.
Without appeal to civil theology in the Lockean manner, it is possible to lay out a case for animal rights in terms of piety. Piety in general is a due-rendering based on some benefit received. Thus religious piety is a rendering of what is due to God as our benefactor; filial piety renders to parents what is due to them in virtue of the benefits received from them; these pieties have extended circles -- for instance, in filial piety there is a justice not just between ourselves and our parents, but a justice between ourselves and the family to which we are related through our parents. Thus our cousins have claims on thus -- weaker than the claims our parents do, but intimately related to those our parents do. To claim that animals, or even the natural world in general, have rights is, in fact, to claim that there is such a thing as natural piety: we render to animals (or to the world, or to what have you) something proportionate to the benefits we receive from them. And not only that, this piety can potentially have an extended circle as well. It is certainly clear that we are benefitted by non-human creatures; the natural question, then, is: "What demands of justice does this therefore establish with regard to them?"
It is along these two lines, I think, that we can establish the claim that animals have at least a weak form of right. They have rights in virtue of being good creatures of God, and in virtue of being our benefactors, in however weak a sense. The problem with basing animal rights on interests is that the only interests that can establish rights are just rights, so an interest-based account, if it is to work at all, simply reduces to a justice-based one.
UPDATE: Lee has a nice response with which I largely agree. I suppose I'm inclined to see interests (and things like the capacity for experience that support them) as having very little role to play in terms of obligations -- and therefore rights -- although they do have a very important role to play in moral taste, which is a different aspect of our moral life. Some things that don't violate rights or obligations are nonetheless in bad moral taste -- they're really not appropriate to a decent and civilized person; likewise, some things that aren't in bad taste morally may nonetheless be forbidden by some obligation, due to circumstances. So I think I'd concede that interests have a role to play in our moral relationship with animals, but deny that this role has much to do with rights; interests are relevant to a different aspect of our moral life.