It seems like this might be a bug with the "Developer Tools" feature of Chrome. The whole time I was trying to set a cookie (but not retrieve it) and it worked with the other browser. It worked, assuming you trust the cookie viewing section of FF or locate the cookie's file for IE. In Chrome I was relying on the "Cookies" section of the "Developers Tools" (Developer Tools > Resources > Cookies).
I decided to take a step further and actually output the cookie's value using this script found in WHT (posted by Natcoweb):
<?php
setcookie('test', 'This is a test', time() + 3600);
if(isset($_COOKIE['test'])){
$cookieSet = 'The cookie is ' . $_COOKIE['test'];
} else {
$cookieSet = 'No cookie has been set';
}
?>
<html>
<head><title>cookie</title></head>
<body>
<?php
echo $cookieSet;
?>
</body>
</html>
And it worked on all browsers including Chrome (I get: "The cookie is This is a test")! However Chrome's cookie inspector continues showing "This site has no cookies". I also managed to find the list of cookies stored in Chrome's settings (Options > Under the Hood > Content Settings > All cookies and site data) and finally found the cookie (more steps to check but at least more accurate than developer tools)!
Conclusion: cookies were being set, but Chrome's development tools can't see it for some reason.