Is it fine to bunch together power wires?

The biggest issue occurs when cables run parallel to each other. Crossing is not so bad, particularly at right angles. You are better off having a jumbled mess than having all the cables neatly organized running parallel, even if they are spaced a 1/2" apart. The amount of current that is induced in the neighboring cable is dependent on how much current is running through each cable, how close together they are, and for what distance. Having cables occasionally cross each other will result in a lot less induced current than running them parallel in close proximity for more than a very short distance. 

XLR cables have good common mode rejection (assuming the signal is really balanced and the receiving end is using a differential input or transformer to cancel out the common mode signals). This means that any induced signal from neighboring cables should be nearly the same on both signals within the cable, and therefore be cancelled out. 

So, to answer your question, it's much less likely to be an issue with balanced XLR cables than with other cables. Also, the current in balanced interconnects is fairly low (assuming reasonably high input impedance on the component you are connecting to), so the amount of coupling is likely to be low.

It's probably not optimal to bundle them tightly together, but recording studios will often run lots of balanced cables through the same conduit without issue.