Display next 3 posts based on custom taxonomy

From what I gather all you need to do is query the 3 posts and step though each one using the_post(). I have not used it like this so not 100% that is how it works.

the_post() Retrieves the next post, sets up the post, sets the ‘in the loop’ property to true.

$project_query = array(
    'posts_per_page' => 3,
    'post_type'      => 'projects',
    'taxonomy'       => 'project-category'
    );

query_posts ( $project_query ); while ( have_posts() ) :  

//Post 1
the_post();
the_title();
the_content();

//Post 2
the_post();
the_title();

//Post 3
the_post();
the_title();

endwhile;