Filing the discharge petition is only the first step. That leaves Republican House members with two choices:
Sign the petition and be beat up by the Tea Party in the primary.
Don't sign the petition and answer to public outrage.
How they will decide depends on how great the public outrage will be. And that depends on how much outrage WE, the readers of Daily Kos and other dedicated Democratic volunteers generate.
If we do enough, we'll win however the vote goes. (For this post, I'm assuming the discharge petition deals with the minimum wage. Most of what I say, however, would apply to a discharge petition for Immigration Reform.) If the petition gets enough signatures, then the minimum wage goes up, tax receipts go up, SNAP payments go down, and so does the deficit -- to speak nothing of what benefits individual recipients and their families get.
If the petition doesn't get enough signatures, and Pete Roskam and [insert the name of your local Republican villain here] don't sign it, then those guys earn public odium.
There are places -- USA Today comes to mind -- where there is a reason to go after "the Republicans." Generally speaking, it would be more productive in most venues to deal with some particular Republican Rep.
How do you get the word out? Any way you can.
A LTE in the local paper is always one way.
If you live in or very near the district, then simple word of mouth is a live option. It is better to light a candle than to curse the darkness, and it's better to tell one person than to seethe in silence.
There is an attempt at hyper-local web pages called Patch. If there is one in one town in the district -- or in each of several towns in the district -- that's another outlet.
Remember that if one R cracks, that builds the pressure on the others. If none of them crack, we hope that it will wound them in the Fall.
On the other hand, if the Democrats file a petition and no Republican suffers for not signing it, the likelihood that any Democrat will file another discharge petition gets much lower.
One of the problems of dealing with individual Republican members of Congress is that most of them haven't taken all that many bad votes. (Repealing the ACA is an exception.) Many of the important issues simply haven't come before the House, which can be spun as Boehner's fault and not the individual's. A discharge petition strips away that fig leaf.