Twentytwelve, Suffusion, Responsive (many others) all work great on mobile devices, including menus. They may not have the exact look or layout you want - but that's generally customizable - though if you change things, you need to take care not to "damage" the responsiveness.
Oops, looks like you're already trying Custom Community which is another nice one - and if you have questions about it, you an ask the developer here: http://wordpress.org/support/theme/custom-community