Well it really depends on your niche on how you promote it. If yours is something along the lines of "kitchen sink" I guess your best bet would be to post them in home improvement forums/G+ pages etc. However I don't think this will get you constant traffic, rather just little boosts that involve too much time. Amazon affiliate websites are more suited to SEO traffic in my experience. So you want to create content people enjoy and want to link to.
I guess I can just tell you want made my website rank and hopefully that'll be some help.
There's two camps of Amazon affiliate marketing when you first start: (1) Intense keyword and competition research, and (2) write about something you enjoy/already have a little expert knowledge in. For my amazon site I went down the "write what you know" route, because in the end the only way to stay in the top pages of google is if your content is killer. AKA Content is King! (cliche, I know).
Secondly I didn't want to constantly be updating my blog as I have university and other commitments. Today I only post once every 2-3months. So the first thing I did was create a resource section prefect for my niche that would be timeless. This section involved no affiliate links, it was just about making great/timeless content people would find useful. This also helps build authority.
After that I created a facebook, and instagram and posted cool images relevant to my niche as well as my own website content. I sort of built a mini brand, for example I have ~4000 facebook likes and ~1500 insta followers. I also found Google+ communities to post in. However you never post just plain "click here for best kitchen sinks" – you want to create value. So you want to first comment and contribute or simply like other people's posts. Then you can post your posts, but don't make it sound like your selling something.
NOW FOR THE AFFILIATE POSTS
This is when I created the "best xxx xxx" and "Best xxx xxx under $xxx" posts. But once again I decided I didn't want to straight up sell something, I wanted to create value as that would stay relevant. So these posts are very detailed explaining the different classes of XXX and what features to look out for, and the pros/con of different XXX. And at the end of the post I have a "MY PICKS SECTION".
Now that you have killer content , you need to promote it. I posted them on my social media posts, G+ communities and also I took the advice of the guy below to build links to them. Pretty much reaching out to influencers etc (I don't do this much as I don't have the time, but I really should):
(1) Create A resource site
(2) Create killer content
(3) Build a mini brand via social media/mailing list
(4) Create keyword researched buyer intent posts that contain value
(5) Promote content via your own social media/relevant forums/email list and email influencers who may be interested in sharing your content.
Hope I helped.