Get multiple term objects by ids

I have an array of ids that correspond to terms. How do I get the full term objects for each id?

$ids = array(1,2,3);

Ideally, I would love to be able to do something similar to the way taxonomy queries with WP_Query work, but obviously with terms instead of posts.

I wonder if you mean something like this modified Codex example:

// Fetch:
$terms = get_terms( 'category', array(
    'include' => array( 1, 2, 3 ),
) );

// Output:
if ( ! empty( $terms ) && ! is_wp_error( $terms ) )
{
     $li = '';
     foreach ( $terms as $term )
     {
       $li .= sprintf( "<li>%s</li>", $term->name );    
     }
     printf( "<ul>%s</ul>", $li );
 }

where $terms contains an array of term objects, empty array or the WP_Error() object.

Check the Codex on get_terms() to get more information on the output and the input arguments. There you can get more code examples.