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I finished my screeners about 3 weeks ago and there are still a handful of firms that I have not heard back from. While most of these (I am sure) are dings, there are a few firms that no one that I know has heard anything from. Should I shoot the recruiter a quick email following up? The basic message being something like "I am still interested in X firm and wanted to follow up from the screening interview"?

I don't see a downside but at the same time, I do not want to appear rude or pushy.

A short follow-up definitely wouldn't hurt in this case, particularly if it's after a couple of weeks. More than one follow-up, or sending an email and making a phone call, or following up after only a couple of days -- those are rude and pushy. A single, short email after two weeks is not.