I assume you care about books and are passionate about them. Even if you get some of them for free, you spend quite a bit of money on the rest.
You also spend time with books. Lots of time. For example, right now I'm reading a considerable amount of phenomenology, especially the work of Jean-luc Marion. Pushing through to comprehending perhaps the most important philosophical school of the 20th century--and its implications for things like theology and preaching--will hopefully result in a breakthrough in my own preaching and pastoral ministry. It is worth the time, both in terms of professional development and pastoral ministry.
This illustrates that, underneath any techniques, the best way to get books for free is to be passionate in your engagement with them.
But I digress. You want to learn how to get your books for free. Here are three books I have received for free for the new year, and plan to review for various publications. The story of how each one was gotten illustrates how you might get free books.
Write for an on-line journal
In this instance, because I've written content for the Journal of Lutheran Ethics in the past, they looked through their stack of books, and asked if I might review this one. Since I write the preaching column for JLE, this makes sense. Find your niche, then work it. Keep your name in front of the editor, and be in regular dialogue with them. That's the angle here.
Write for a print journal
Most journals can request review copies for you if they haven't already assigned the book to somebody else. In this case, I requested the book from the Book Review Editor, who wrote to the publisher. The publisher is doing a special printing of the book and sending it my way (apparently the print run is so small they are printing it on demand). Again, direct contact with the editors, and an established relationship with them, is essential.
Write to the author
This book is published at an institutional-buyer-only price-point. I doubt anyone other than libraries will purchase it, unless and until it sells enough copies for them to consider publishing it in soft cover. In cloth, it is jaw-droppingly expensive. You can actually read it on the publisher's web site as an e-book, but the subscription for that service is even more jaw-droppingly expensive.
Or you can just write to the author and see if they have a copy lying around to send. Which Professor Dorrien did, so is sending one my way for review here on the blog. Since the topic is of interest as philosophical/theological genealogy leading up to phenomenology (the topic I mentioned at the beginning of this post), the book was available because of a) my interest, b) my connection to the topic, and c) because I was willing to review it.
Almost any author you can think of has a web site, a faculty page at their institution, or is on Facebook. Make the connection. Write to them. The world is flat. These folks don't live on Laputa.