A few years ago if you told someone that higher dimensions could be detected with current technology they would certainly look you as a crackpot. The idea of additional dimensions is older than you might think, it actually goes back to the work of Kaluza and Klein (in the 1920's,) who discovered that if we introduce and additional spacial dimension curled in a really small radius to general relativity the resulting set of equations not only included the Einstein field equations, but also Maxwell equations!!! Despite it eventually failed as a unified theory of gravity and electromagnetism the idea was explored again when theories of supergravity suggested that in 11 dimensions the unification of all known forces could be possible. In the early eighties it was clear that supergravity wasn't the answer to unification (it was not renormalizable), but a few years later the famous result by Schwarz and Green that superstring theories in 10 dimensions were free of anomalies, anyway by the beginning of the 1990's the field was starting to stagnate, until the work of Polchinsky and Witten showed that the different string theories were only different descriptions of a single theory and also that 11 dimension supergravity was a low energy approximation!!! During this time the additional dimensions still were considered as small and curled, avoiding any realistic attempt of detection.
The announcement that 10-D string theories and 11-D supergravity were related was considered the second string revolution, in this revolution not only strings but also new objects called branes came into spotlight. The branes lead to to the Randall-Sundrum models, in this models our Universe is a five-dimensional space and the elementary particles except for the graviton are localized on a (3 + 1)-dimensional brane or branes. Since this models were proposed the detection of higher dimensions seems possible.
The approach is to create a high energy graviton which could leak in to the extra dimensions and disappear. Of course we do not detect directly the graviton, but we can see that energy and momentum were carried by some invisible particle. So this is the simplest kind of experiment you can do, and if you can eliminate other kinds of possibilities for things that carry off energy invisibly, you would then be able to claim that you've seen evidence for extra dimensions of space. Let's be honest, rather than a test of M-Theory this is really a test of large additional dimensions and not of M-Theory itself (we don't even know it's dynamics), anyway the Randall-Sundrum models are clearly the products of some ideas introduced in the second string revolution and any experimental evidence of them will clearly show we are in the right track.
You can see a technical account here by Joe Lykken. Anyway you should note that the real chance of detecting this additional dimensions in the LHC is very small, even in the proposed ILC that should allow us to do better measurements in this issue the chances are scarce, but the nice thing is that we have at least a chance to do an experimental confirmation of the this models.
By the way, if you are on the Loop Quantum Gravity camp, there are also some experimental predictions (concerning light propagation) which I will comment in a future post that are actually accessible and can definitely verify or falsify the theory .