What methods do you guys use to give away free reports?

Free Reports or not, you still need to drive the traffic to give them away. You can drive traffic with any methods and you've touched on article marketing. That is just one way.

Free reports have to part of an overall game plan. For example, the free report can be the give away to build a list that will then have a sequence of products that you will market to that lead. This way, you can afford to pay for traffic to the free report, knowing very well that your backend system will pay back in many folds.

Hope this short scenario helps.

That's not article marketing. That's article directory marketing.
Because of their lack of quality control and refusal to block the
cheaters and the spammers, article directories are no longer ranking
well in the search engines.

That doesn't mean article marketing doesn't work. It just means that
you have to go back to doing it the old-fashioned (better) way, and
put your articles in places where people still read them.

Contact publishers directly, build a relationship with them in which
they can expect the kind of articles from you that they like to publish.
There is nothing new about this and you won't have to worry about
changes in Google... This was being done long before the Internet itself.

You'll be building two lists:
1. Publishers
2. Prospects

Good luck!

Create an optin form with a strong headline using the reports as a benefit for optin in!

You can use these free reports to drive traffic to your website or blog by submitting them to websites with free ebooks, but you won't be able to collect email addresses and build your email list this way.

Article marketing is better than ever, Robin.

What you used to use was article directory marketing, which was never intended to be a traffic source in the first place (but could be used that way, at one time), and it's article directory marketing that's "not as good as it once was".

It did ... well, to be fair, using article directories "as a traffic source" was only ever based on a misunderstanding of their actual purpose, though I hear that before about 2009 it did work, to some extent.

Article marketing works better than ever.

I've also heard of people attracting large numbers of downloads of "free reports" they want to circulate by putting them in the black-hat, file-sharing and torrent sites themselves, and allowing people to believe that they may be illicitly downloading "pirated copies" of something. That appeals to a certain type of person (and there are large numbers of them). Whether those people will then turn into good customers for you, or not, I have no idea.

For myself, I'd try to use them for list-building (because I happen to be an affiliate marketer, and to all realistic intents and purposes there's no money in affiliate marketing without list-building), so rather than just giving them away free, I'd be using them as the incentive on opt-in pages, and drawing traffic to the opt-in pages (which is a big part of how I make my living anyway).

I would (and do) use article marketing for that, though I'll consider almost any other kind of traffic, too (apart from intentionally targeting SEO traffic, which - for me, at least - is crap traffic and a waste of time).

Yes ... I think your question really boils down to "How do you get traffic?", doesn't it?

There are - literally - thousands of threads here addressing that question. Using the "advanced search" function to find Main Marketing Forum (only) threads with the word "traffic" in their titles (only) will pull up the most recent 500 such threads.