Would it be possible for a fully modern civilization (i.e. quite like our own, perhaps slightly more advanced, with near-future tech) to arise on an Earth-like planet devoid of uranium, plutonium, thorium, or any other elements that could lead to the discovery/development of nuclear power or nuclear weapons?
When you really look at it, it seems most technologies present today don't take advantage of nuclear power or radioactivity. Sure, smoke detectors will not exist as they do here. There will be a part of their energy sector equal to about 11% that requires some other solution than the one we have found. In any case, it seems very little in the way of civilization has to do with nuclear weapons, radioactivity, and radioactive isotopes.
In short, a societies' infrastructure, medical care, food supplies, and popular philosophies will generally not care if every nuclear isotope was removed from earth, and will get along just fine. Whatever they do use nuclear radiation for, however, can be replaced. If those areas of expertise are not what you measure civilization by, I'm not sure how else to do it.
One can argue that their version of Japan would still fight tooth and nail in their WWII because no show of conventional force would convince them to surrender. One could also then argue that there would be no MAD doctrine which came out of their Cold War. One could argue that the absence of nuclear weapons would have help thaw the Cold War; we could at least fight each other like modern humans and kill each other off until the non-involved nations were suddenly the powerhouses. So yes, radioactivity had a decaying hand in our history, but many of the major technologies and philosophies we have which separate us from humanity past hardly rely on nuclear anything.
In short, these other people would also be missing out in the other two fundamental forces of the Universe, and some things wouldn't make sense. I see no reason why they wouldn't have similar technology and be just as "civilized" as we are. ("Civilized" being whatever measure you take to them except things directly related to nuclear activity!)
But Will They Discover Radiation?
They simply need to figure out what is powering the sun. Some bizarre thing called "new-clee-air fuze-on"...
Really, though, there are plenty of isotopes of elements which are not heavy metals but are radioactive. (Such as the popular C-14!) Assuming none of these are present on this particular planet, that will make their version of the Curies happen somewhat later in their development. I predict they encounter radioactivity when they start sending things into space. At the very least, they'll figure it out when they see stars producing heavy elements from lighter ones.
Interestingly, I remember a short story, maybe written by Robert Sheckley (have to ask on the sci-fi sister site for identification, I think) where aliens actually stopped all sorts of radioactivity and nuclear reactions on earth. Unstable isotopes were rendered stable, no radiation, and so on and so forth.
Secondly, said aliens removed all knowledge about radioactivity and nuclear physics from earth, too, except from a select few.
While the overall changes in this world have been touched briefly, the biggest change was actually that one intended by said aliens, to deprive the humans of their weapons of mass destruction.
While said world wouldn't have any nukes and the energy sector would be moving from fossil fuels directly to renewable energy sources - if they can commit themselves to the change - and applications of isotopes like medicine would be unthinkable, there is one bigger issue:
Vital parts of modern technology are based on quantum mechanics. Starting with simple LEDs, continuing with lasers up to any computer chip, many appliances vital to our living are developed on the knowledge of quantum states and quantum theory and its development was heavily influenced and spurred on by nuclear physics.
So, in essence, without any radioactive isotopes and nuclear reactions to study it would be very difficult to build the groundworks of modern or near-future civilization and maybe some handwaving is required for your people to pass over this lack of knowledge and develop their own version of quantum physics.