Alt Text
 

So I've been putting alt text on each of my full-size pix. Should I be putting it on thumbnails also?

And...how much is considered keyword spamming?

As long as you don't use the same keywords on all thumbs and as long as your tags don't have 20 words each there should be no problem and yes you should use them, not only for seo purpose. It certainly can help with Google image and generic traffic but without inbound links you it won't do nothing to your SERP ranking.

Describe and insert the right keywords in each thumb. and as DirtyProfits ses, inbound links is much important alt tags is just a part of many factors to be good on SERPs.