They are opposed by victims rights groups; the vast majority of states, which have a patchwork of laws granting DNA access; and the federal government. The governments say that creating a constitutional right to the testing would infringe on states' rights, overwhelm them with frivolous demands and create an endless right of appeal for those convicted of the most violent crimes.
Set aside the states' right argument for now. I'm more interested in the question of whether constitutional protections for DNA testing would, in fact, result in lots of frivolous demands and endless appeals.
If there were, literally, no restrictions at all, maybe that's what would happen. Maybe every con with time on his hands would demand test after test just for the hell of it. Maybe. But if the court required even a minimal showing of cause, wouldn't frivolous requests dry up? What's the point, after all? If you're guilty, then you know perfectly well that DNA isn't going to get you off the hook. So why bother?
That's why I've never found this argument very persuasive. Prisoners who know they're guilty have little incentive to demand DNA tests. Conversely, though, prosecutors have loads of incentive to deny DNA tests, even — or maybe especially — in cases where it might well prove wrongful conviction. This suggests that the court should adopt some kind of balancing test: not an absolute right to endless DNA testing, but at least a presumption in favor of it. Make the hurdle just high enough to deter the genuinely frivolous, but low enough that nobody has to rot in prison for years just because they didn't have access to a simple test. We are, after all, in favor of not imprisoning innocent people. Right?