Well, in my case there were a couple of boxes to check off.

First, find the right person to approach. Papers and sites often have an email for people to pitch their article ideas to - don't use those, they're often not well read or, in my case, the person reading it probably can't / won't judge your piece for what it's worth. Find a journalist or person who writes about the same thing you write about, or someone who know his / her shit. They're already interested in your topic, so they're easier to convince to read your writeup.

Second, write something that you stand behind completely. Not just because you agree with what you say - because that's almost always the case - but also because the facts check out. Use multiple sources, do your research well. Make sure that your piece makes a strong point. This is necessary because you have to convince that journalist that your piece is good.

Third, keep it short. In emails as well as your piece - you can always expand on ideas or explanations later, when the editing process starts. Kill your darlings, as they say.

If you've got more questions, feel free to ask.