Amazon lets anyone host a giveaway of any print book sold by Amazon. If you look on the product page of a book, you'll see the icon right below the "Create a review" button.
Interestingly, you don't have to be the author or publisher to set up a giveaway. The link appears for everyone. Once you click it, you have to commit to paying for the books you're going to give away, select the number of copies, the odds of winning (e.g. one in 100), and provide the wording for win and lose messages. You can also specify a single requirement, such as making people follow you on Twitter or watch a short video or follow the author on Amazon. You can even control whether the link can be shared or not, which controls who can see it.
If you wanted to find a way to gain more Twitter followers, for example, you could pick a popular book and host a giveaway of it. This is more feasible now, because Amazon now allows folks to host a giveaway of a Kindle book. This is much less expensive (cheaper price and no shipping costs), so it's easier to give away more copies.
I did some print giveaways a while ago, and now I've started one for the Kindle version of Saronna's Gift. Click here to enter!
nb: The link above will only work until the giveaway runs it course and all five copies are claimed, or March 16, 2016, whichever comes first.