The main reason I want the US to avoid building a bunch of nuclear power plants is because of this:
http://en.wikipedia.org/wiki/Chernobyl_disaster
Also, where does all the nuclear waste go?
Underground in barrels that will contaminate our water supplies. 
Hate to tell you this, but "Chernobyl" is the weakest argument possible when arguing against nuclear power plants. The waste issue is a legitimate argument, but Chernobyl and Three Mile Island (TMI) are not.
Chernobyl was analagous to having an arc-welding business in a building made out of dynamite. It was only a matter of time before it went. The U.S.S.R. knew about it, the entire world knew about it, and every nuclear agency on the planet knew that it was a time bomb. The design of the plant was such that the U.S.S.R. could extract the plutonium from the spent fuel while keeping the reactor running thus giving them a continuous supply of plutonium for their weapons manufacturing. There was no containment for the reactor and the night of the accident all safety systems were shut down to see how long they could keep the reactor going in the case of a power loss. The problem is that the "experiment" was conducted over mulitiple shifts and some crazily stupid things that were done by the people in the first ship weren't fully told to those in the second shift. The physical design of the reactor was also a design deemed inherently unsafe due to the materials it was constructed out of, and once the nuclear reaction inside starting getting out of control there was no way to stop it. The core completely melted, the hydrogen gas buildup inside the core caused a massive explosion, and that entire part of the world was forever doomed thanks to the U.S.S.R.'s greater concern of producing weapons grade plutonium rather than safe energy.
The design of the Chernobyl power plant is a design that is not allowed by any regulatory agency. One reason is that it allows for mass production of plutonium, and the other is that it is inherently unsafe. The TMI incident that occured here in the US was in a pressurized-water-reactor (PWR) which, by design, has an immense number of safeguards. The reactor core itself is setup so that in the case of a system failure, the control rods which absorb neutrons and stop a nuclear reaction will smother the core. (This wasn't the case in the Chernobyl incident). In addition, the TMI reactor's safety systems are set up in such a way that if there is an issue with the safety system, the reactor is shut down. Nuclear Reactors in the US are surrounded by a MASSIVE steel and concrete containment shell which has been tested to be impervious to missile attacks. In the case of an ultimate catastrophe of every safety system in the design, the containment building keeps everything inside.
At TMI, there was no measurable increase in radioactivity outside of the reactor building. The incident resulted in damage to the reactor core that could not be repaired safely, but the area wasn't flooded with radioactivity and in fact the other reactors at the plant were in action until about 10 years ago. The designs of power plants not designed for plutonium production are incredibly safe.
Now as for the nuclear waste, that is an issue. As nuclear power plant designs get better, the waste of nuclear fuel becomes less and less. Still, you are going to have a great deal of nuclear waste that you can't do anythign with for hundreds of years because the daughter products of nuclear fission retain their intense radioactivity for quite some time. (Uranium itself, while radioactive, isn't all that bad. It's half-life is so long and it's radiation is so weak that it's not a threat at all. It's the daughter products like radon, cesium-137, etc. etc. that are immensely radioactive in terms of activity and energy. The waste can't be processed until those products have reduced).
I believe that France has close to 80% of it's electricity generated through nuclear power plants. It'd be interesting to see what they do with their waste.
Here in the USA, you will never see nuclear power gain acceptance. Too many people just aren't bright enough to realize that radioactivity and nuclear power aren't things to fear, but things to respect. VERY few people know that the accident at Chernobyl simply is not physically possible here in the USA with the plant designs we have. They also have this internal fear (thank you mass media) of anything radioactive. Christ, I go home every day and sitting in one of my cabinets is 10 grams of pure uranium metal. It's not that radioactive. (Granted, it is sitting inside a thick walled lead can as I do respect radioactivity, and it makes a geiger counter go crazy when it's near it, but the energy given off is quite weak). You get FAAAAAAAAAAAAAAAAAAAAAAAAAAAAR more radiation exposure from being in a plane than I do from my uranium. (The thin walled aluminum shell of the plane provides little to no protection from the cosmic rays bombarding down on the earth that you typically don't get exposed to).
Anyway, I think nuclear power is a good way to generate electricity (and it's fairly cheap. My electricity is generated from nuclear power and the electric bill has never been a problem for me), but until the common public can grow up a bit and learn something about nuclear power, and we can find a proper method for dealing with the waste, we'll never see it here in the US.