The Bohr atom, in structure, is not very different from the Rutherford atom - you have an electron going around the nucleus in a circular orbit. So even in the Bohr model, the electron must experience some centripetal acceleration. Why is there no radiation in that case, like in the Rutherford's model?
The early 1900's are considered revolutionary in physics because for one of the first times, we were discovering that physical laws could be very different in various different domains. What does that mean?
Take for example, a mass m moving with speed v will have a linear momentum p=mv, as we all know; but if the speed v is very high, close to the speed of light c, according to Einstein's theory of special relativity, the momentum will be p=mv1−v2c2, which is obviously not the same as mv. As we go from the domain of slow moving objects to the domain of fast moving objects, we encounter the new physics of special relativity, where the laws of the previous domain do not apply, and the physical laws go through an extensive revision, sometimes complete changes.The pioneers of the modern atomic models faced a similar paradigm shift while trying to model the atom based on the 1909 gold-foil experiment. Bohr argued, that at small length scales, classical physics (Newtonian mechanics and Maxwellian electromagnetism) that we are so used to, breaks down, and an entirely new set of laws overtake. This domain of the microscopic world is the domain of quantum mechanics, and you simply cannot work with classical laws at that level.
So basically, an electron does not radiate energy in Bohr's stationary orbits simply because Maxwell's electromagnetism does not work in the same way at the length scale of an atomic radius, as it does at the macroscopic scales that we are familiar with.