How do ascending neural pathways filter unimportant information?
A neuron cannot know the importance of a signal in any applied sense, that is, it can't know the difference between one triggered by a feather or one by a hammer... on an individual basis.

A single neuron can however accumulate information in a number of ways, either by requiring multiple incoming triggers (filtering out localised signals, i.e. from a single connected neuron) or they can require repeated triggers (filtering out weak/freak/accidental signals from a connected neurons). This is called summation: 

These filtering events occur at the synapses, so how would a neuron filter a signal as it was passing through it? As far as I know, it doesn't!

NOTE:
As noted by @WSYWYG, LTP and LTD are also important, they stand for Long Term Potentiation and Depression. These are names given to the long term increase or decrease in sensitivity of a synapse, that is a neuron can become more or less likely to pass on a signal over long periods of time due to the pattern or frequency of signals it receives. Googling the terms will lead you to a wealth of vastly more detailed information on these topics. 

EDIT:
As an addendum, your body can inhibit neurons selectively, so as a wild stab in the dark, it may be interesting to see if ignoring a sensation from part of your body is the direct result of of external inhibiting (filtering) of signals incoming from ascending neural pathways... but don't trust me on that!