The Twitter API doesn't support username and password anymore. They move to OAuth exclusively a couple years ago, but only some of the API's required authentication - that was v1.0. Recently, the Twitter API introduced v1.1, which LINQ to Twitter supports. The Twitter API v1.1 requires OAuth on all queries and commands. So, LINQ to Twitter is supporting the type of authentication as the Twitter API.
That said, here's a couple tips that might make your user's experience much better. You would include your application's ConsumerKey and ConsumerSecret, which won't change, but don't require the user to obtain separate ConsumerSecret/ConsumerKey pairs. In fact, this is the way OAuth is intended to work.
Another tip is that Twitter authentication tokens don't expire for a user. This means you can ask the user to authenticate one time, during sign-up, and then save their credentials after they're authenticated. The credentials are available via the authorizer you used to do the OAuth process. After that, just re-use the saved credentials, loading all four credentials into the authorizer and LINQ to Twitter will just do (signed) queries without making the user authenticate again.
BTW, I watch the linq-to-twitter tag here on SO and have a CodePlex.com discussion forum. I'm on Twitter too, but sometimes answers like this are challenging via a tweet - though I have been known to tweet an answer or two. :)
@JoeMayo