I hadn't heard of it before, so this analysis is based on a few minutes of reading some google searches, but I'm already finding major holes.
The cannon of neuroscience is that the electrical activity of each individual neuron sums to form the global EM field. Researches who use EEG (electroencephalogram) certainly use this fact to gain information about the brain.
The CEMI theory seems to turn this on it's head and say that the global EM field is controlled by 'free will' and can also effect the firing of individual neurons.
Unfortunately, the sum of evidence seems to support only the neuron -> global field rather than the global field -> neuron in all but the most extreme circumstances.
Things change your EM global field all of the time and don't manage to change consciousness. Cell phones radio, and pretty much every electrical appliance have some small effect on your em field, and don't seem to effect consciousness in any strong way. That's the first knock against it.
In some extreme circumstances, though the "global field" CAN effect neurons firing. We can stick an electrode very close to a neuron and induce a large EM pulse and get a neuron to fire that way, but we must be very close. At that small scale, the function of the EM field and what a neuron already does (voltage sensitive ion channels which propagate action potentials) is essentially the same, so reducing the CEMI theory to that level would render it meaningless.
Another circumstance where the global field can effect single neurons firing is TMS or trans-cranial magnetic stimulation. Here a metal coil (or usually a figure 8) can stimulate or suppress the firing of large groups of neurons up to centimeters away. However we run into a couple of problems. First EM fields are fairly uniform. That's another way of saying that if you want to produce a field that can make a particular neuron fire or stop firing, you need to do essentially the same thing to the neuron right next to it. This type of crude control over large groups of neurons isn't sufficient to really control consciousness in the complexity we understand it now where many closely spaced neurons may perform drastically different roles. The second and perhaps more convincing reason against this type of control is that it takes a huge amount of power to cause large voltage shifts in neurons centimeters away through bone and tissue. I mentioned EEG above. In those type of experiments the strength of the global field is about 3 orders of magnitude smaller than could drive the action of a single neuron, so there is no evidence that there are global field fluctuations that are large enough to modify the firing of a single neuron at any significant distance.
The real nail in the coffin of CEMI though would have to be the first law of thermodynamics. Regardless of the spatial scale, modifying the global EM field requires energy, but there is no known source for this energy.
So across the board, it fails the test of an alternative hypothesis in that it does not describe what we already know better (or even nearly as well) as the current theory.
Hope that was coherent.