Is routing signal traces under components a good idea
The answer as always is: it depends.

In general, if the PCB design rules allow it (i.e. you are not violating minimum spacing, etc.), you should be OK. There are a few other reasons to avoid doing so, however:


The signals are high-speed signals that can be sensitive to coupling from any transient current in the components they are routed under. For example, if your trace is a USB 3.0 signal, you don't want to route it under the decoupling capacitor of the output of a switching power regulator.
The traces are high-impedance nodes in your circuit that can easily pick up charge from adjacent nodes through parasitic capacitances between traces. For example, if the trace is the reference voltage used at the input of a comparator, then you probably want as much shielding and isolation from other traces on your PCB.
If you anticipate the need to rework the components you might run into a risk of damaging the trace depending on how good your reworking skills are :). This is more of a practical issue to consider.


But if your signal and the component terminals are not sensitive nodes (i.e. a CMOS digital output under a decoupling capacitor), then you are probably fine.