My friend Brightblades is right in one thing. It seems your teacher was working off a caricature of what the theory of evolution actually says. First of all, you should read Sklivvz's excellent answer at this question. Now to address the elephant in the room, the accident at Chernobyl only happened in 1986. That was only 26 years ago. In that timeframe, noticeable effects in an animal population really would not be at all noticeable. Furthermore, the paper cited by Marta Cz-C actually shows that there have been some changes (in fungi though, not animals).
fungi seem to interact with the ionizing radiation differently from other Earth’s inhabitants. Recent data show that melanized fungal species like those from Chernobyl’s reactor respond to ionizing radiation with enhanced growth. Fungi colonize space stations and adapt morphologically to extreme conditions. Radiation exposure causes upregulation of many key genes, and an inducible microhomology-mediated recombination pathway could be a potential mechanism of adaptive evolution in eukaryotes.
Read the rest of the paper for more information on how there have been some other slight changes to fungi at Chernobyl as well as other locations throughout the world.
Now I am going to repeat a bunch of stuff from one of my web pages that talks about evolution. This web page is set up mostly to deal with creationist arguments, however, the caricature is so severe as to warrant this. As I said earlier, evolution is a population phenomenon.
Evolution acts upon heritable variation of characteristics, and you can only have variation of this sort within a population. A single individual organism, at least if its a multicellular eukaryote, has a fixed genome. It can't change what it has inherited. But a large number of organisms can all have different genomes, and can disseminate variation via inheritance to the next generation. It is upon the population as a whole that evolution acts, with various mechanisms coming into play to remove some variations from the population, and propel other variations to numerical dominance within the population. The organisms in question remain part of that population, and within a generation, those organisms don't change. But the moment a new generation is produced, dissemination of variation can result in the appearance of a new feature in one or more members of that population. If that new feature leads to greater reproductive success for the organism possessing it, that feature spreads through the population, as more and more future offspring inherit it. Over time, the population changes, and more and more organisms with new features appear within that population.
Understanding inheritance basics and mechanisms for changes (genes), we have all that is needed for the appearance of cladogenesis events. Split a decent sized population of living organisms into two, and let's call these new, separate populations A and B. Now let a barrier be erected between population A and population B, so that individuals from one cannot reproduce with individuals from the other. This barrier can be an insurmountable physical obstacle, for example, but this need not be the only form such a barrier can take. Now, first of all, there is no reason whatsoever to think that population A and population B will start off in identical states to begin with. After all, those two populations were derived from an original population comprising lots of organisms with different genomes, and the likelihood of population A and population B being identical at the start of this process is vanishingly small. Then, once our barrier is erected, and our populations are allowed to reproduce separately from that point on, there is no reason to think that those populations will move in the same direction in the long term. Indeed, it is far more likely that they will be subject to different environmental and ecosystem influences, and those different environmental and ecosystem influences will shape the long term heredity of those populations. Indeed, that's all that natural selection IS - it's a single, concise term used to encapsulate all of those environmental and ecosystem influences succinctly, and additionally to encapsulate the fact that those influences affect the inheritance of characteristics within a population over the long term.
As a consequence, any two separated populations of living organisms, that originated from a single population, will diverge from each other. If the extant influences on those two populations are sufficiently different, that divergence will take place more rapidly. Eventually, we will arrive at a point where those two populations become sufficiently diverged from each other that individuals from population A can no longer produce viable offspring with individuals from population B, and vice versa. When this happens, we have a speciation event. Indeed, this has been observed taking place in the wild AND in the laboratory, and has been documented in the relevant scientific papers. So, if anyone wishes to assert that there are 'magic barriers' to speciation or other cladogenesis events, then reality doesn't agree.
You will not have any animals giving birth to any radically different animals as a result of radiation. Most changed sue to radiation will not provide any particular advantage to an animal anyway. Also, a change may also be dependent on a previous change and require many generations to fully manifest. All this was demonstrated by the long term evolution experiment led by Richard Lenski at Michigan State University.
So in essence, your teacher was just plain wrong. As for the findings of the past 20 years, we are always learning more and more. For instance, there has been an explosive capability in DNA analysis and sequencing, which has only provided more support for the theory of evolution. A mechanism that Charles Darwin could not have had any idea about in his day and age, yet it perfectly supports his conclusions. Again, read the answer provided at Skeptics.