%menu_values = {'eenie'=>'one', 'meenie'=>'two', 'minie'=>'three'};

You have the wrong kind of brackets in that assignment. Hashes are initialised from lists, so you should have ( ... ) (round brackets), not { ... } (curly brackets).

If you had use warnings in your code, then you would have seen the error "Reference found where even-sized list expected" - which might have been helpful to you.

Oh, and then you need to pass a _reference_ to the hash into the function.

print $CGI->popup_menu(-name => 'menu_name',
                       -values => \%menu_values);
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