There are numerous implications of choosing POST or GET.
First off, the HTTP specification requires that GET (and HEAD) requests be idempotent^W EDIT: side effect free to enable caching. Clearly, violations of this abound; however, proxy servers and other infrastructure will often hold you to this rule, so make sure you know what you're doing. (proxies and clients are permitted to withhold or arbitrarily repeat GET requests without changing the outcome)
Second, security: although AJAX in the strictest sense falls under the same-origin policy, you can quite easily make GET requests to foreign hosts in JavaScript by creating SCRIPT, IMG or IFRAME elements with appropriate SRC attributes, whereas you'll struggle to create a foreign-host POST. Inappropriate use of GET is therefore the enabler of XSRF attacks. CAUTION: read petewarden's reply regarding XSRF attacks via POST forms.
Third, data size, as others have said. URLs are notoriously problematic for conveying any information other than short text.
Fourth, if web crawlers get hold of any URLs which exhibit side effects on GET, they will wreak havoc on your site. (though this can admittedly be alleviated by judicious use of robots.txt)
Finally, you seem to imply that POST requests will not or cannot generate a response similar to GET. This is not the case, responses to POST work in the same way. (except it must not be cached) The deciding factor is the nature of the request (side effect or no side effect), not the nature of the response.