First of all, before we can talk about whether gods exist, we have to define the meaning of the word, or at least characterize it by some minimal set of attributes. Obviously, if you say that a god is anything that anyone worships, then gods do exist (as idols, for instance). This is not a useful definition in our context.
Many people claim that god is omnipotent, omniscient, and omnibenevolent. In my opinion, we have very good evidence that this kind of god does not exist. If it did, there would be a lot less suffering in the world. Most theists object on the grounds that either (a) we deserve to suffer because we are sinful, or (b) the suffering is for our own good -- a learning experience (with the "benefits" usually reserved for the afterlife). The first objection is easily shown to be false by the fact that innocent children and animals suffer. The second is a little trickier, but I think the enormous amount of apparently gratuitous suffering in the world is strong evidence against the idea. What exactly are we supposed to learn from watching a loved one with Alzheimer's slowly lose his dignity and identity? Or from the existence of parasites which consume their host from the inside out, carefully leaving the vital organs for last.
In fact just the idea of omnipotence by itself is problematical. Can God make a rock so heavy that he can't lift it? Most theists get around this by saying that God can only do what is logically possible, but to my mind this means that omnipotence is an ill-defined word. Certainly Christians tend to claim that their God can do things which on closer examination seem to fall into the heavy rock category. For instance, God supposedly created beings with free will, and yet he somehow maintains control of the universe. If you pray for something (and truly believe, and it's not something bad) God is supposed to arrange things so it comes to pass, but with free will in the picture, it seems impossible to arrange this, even with perfect foreknowledge.
What about a god that doesn't have the big three omni-attributes? Well, my own personal minimalist definition of god is an intelligent being that is not subject to physical law or any analog thereof. If there were an eternally existent intelligent being that created our universe and designed our laws of physics, but was itself subject to some kind of physical laws, then I would not call that being a god.
In science we make the hypothesis that the universe behaves with regularity; that there is a set of physical laws which are true in all space and all time. (I believe Newton was the first person to come up with the idea that the planets were subject to the same laws as matter on Earth.) This idea has held up under extraordinary scrutiny, from the reactions that occur in very distant stars to the events during the first milliseconds of the Big Bang. Many scientists have speculated that other universes might exist with different physical laws, but if this were shown to be the case, they would immediately start looking for some more abstract set of laws which would be true for all universes.
The observed regularity of the universe is evidence (though not proof) that if a creator did exist, he would be subject to some kind of physical law, and therefore he would not be a god by my definition.
What about a tricky god that for some reason does not want us to have any evidence that he exists? While such a god is not logically impossible, I don't take this any more seriously than solipsism. Neither idea is useful for making predictions or even for explaining known phenomena, so Occam's Razor eliminates them.
Bear in mind that proof in the mathematical sense cannot be obtained for almost any fact of daily life which we take for granted. What we strive for is evidence. And in my opinion the evidence is overwhelmingly against the existence of gods.