We have a ton of alternate domain names. What is the best strategy as far as SEO strategy is concerned? Do we want to 301 redirect them all to our primary domain or set up simple landing pages that link to the main site?
There is no SEO benefits of having those domains. You'll want to do a 301 redirects to the main domain so if someone uses one of the alternate domains they will find the main domain. You want only one domain to serve your content as you will run into duplicate content issues with the search engines if you don't. Plus you run the risk of confusing users if they keep seeing different domains when they reach your content.
Having landing pages that link to your site may be viewed as a link scheme by the search engines and you would run the risk of penalties including removal from Google's search results. Even if you don't, forcing users to view those landing pages and having to click on a link to find the content they were looking for will almost certainly result in site abandonment.
To summerize, pick a main domain and do 301 redirects with all of the alternate domains. You'll avoid SEO issues and make it easier for your users.