If a charge travels in a circle it must accelerate, thereby producing EM. However, a wire in a circular loop is analogous to many charges moving in a circle. So, why don't circular currents produce EM? (I have not found any evidence that circular currents can produce EM).
A similar question is: Why doesn't alternating current produce light while a vibrating single particle with a charge will. However, my question asks about circular wires and direct current.
Circular currents do produce EM, and indeed this is exactly how X-rays are produced by synchotrons such as the (sadly now defunct) synchotron radiation source at Daresbury. In this case the current is flowing in a vacuum not in a wire, but the principle is the same.
Current flowing in loops of wire don't produce radiation in everyday life because the acceleration is so small. The electrons are moving at the drift velocity, which is only around a metre per second, so the amount of radiation released is immeasurably small. Synchotrons produce radiation because the electrons are moving at almost the speed of light.
If you consider that electric current is actually the flow of individual charged electrons, then as John Rennie pointed out, the radiation exists but is negligibly small. But if you were to imagine breaking the current into more and more point particles with less and less charge while holding the linear charge density $\lambda$ fixed, then the radiation would decrease more and more, because the radiation per particle decreases as $q^2$ while the number of charges only increases as $1/q$. So in the actual continuum limit where the current becomes perfectly steady, the radiation actually vanishes completely.
To back up John Rennie's answer, consider the Bremsstrahlung formula for velocity perpendicular to acceleration: $P= {{q^2a^2\gamma^4}\over{6\pi\epsilon_0c^3}}$. For all practical purposes $\gamma=1$, so we can simplify this to $P\approx ({q \over \mathrm{C}})^2 ({a\over \mathrm{m/s^2}})^2 {1\over{18.85\times 8.85\times 10^{-12}\times 2.7\times 10^{25}}}\mathrm{W}\approx ({q \over \mathrm{C}})^2 ({a\over \mathrm{m/s^2}})^2\times2.22\times 10^{-16}\mathrm{W}$.
According to Maxwell equations, steady currents and steady charge density won't produce EM waves. So if you have a steady current loop, you can calculate its magnetic field just by the Biot-Savart law, which gives an steady magnetic field in space. If a current loop would radiate energy, then it would be impossible to produce persistent currents in superconductor loops which are actually experimentally observed.(Actually there is also the quantum mechanics in play, and using "only" the classical Maxwell theory isn't completely correct, but it gives you the idea)
But to explain your point about a loop current being charges moving in circle, you are partially correct. If you break the current into tiny elements and calculate the electromagnetic field of each individual element, surely you see that the EM field of each individual element is a propagating magnetic field, but when you sum the EM fields of various elements together using superposition principle, the resulting "net electro magnetic field" would be the same as the one that is given by the Biot-Savart law. (Somehow the same that happen in interference experiments.)
But there is two points to make; Although the steady current and zero charge density is theoretically presumable, this could not be the case in the real world. Since the currents are moving electrons you expect that if you zoom in enough, you see single electrons with some gaps between them, So it seems that in that scale the charge density isn't steady in time and the continuous approximation of charge density breaks in that scale.(Although the currents are produced by electrons in the Bloch states which are not localized in space, So the moving single electrons picture isn't entirely correct either). And if you have a time-varying charge density, you would get EM radiation. Actually this is the situation in the synchrotrons. There you don't have an steady uniform current around the loop, but a bunch of individual electrons that are circulating the loop and you can see the time varying charge density easily. You just can't approximate the electrons running through the loop by a steady current.
