Already using an alternate template?Browser Info Environment
If you don't want to do any coding whatsoever, the simple answer is to create a stack containing the block with its alternate template, then load the stack using the Blocks by AJAX template for Universal Content Pullerhttp://www.concrete5.org/marketplace/addons/universal-content-pulle...
or the Stack Randomizer blockhttp://www.concrete5.org/marketplace/addons/stack-randomizer/,... as described in the documentation.
A route that involves some minimal coding is to create an ajax version of the alternate template, as described in the documentation. You could call it something like 'alternate_template_name_by_ajax', just to remind you of which template to use in the Concrete Custom Templates selection.
The Universal Content Puller addon also includes pluggable wrappers, a nifty way to add additional styling and layout borrowed from other block templates.