(Note that this isn't quite a counter-argument on the same terms. CalmSea's write-up seems to work within the idea of picking a version utilitarianism if one is required to choose utilitarianism but doesn't necessarily agree with its basic argument of always optimizing happiness for any given choice. This write-up will assume somewhat the opposite, i.e. it's an argument against rule-based utilitarianism assuming that the basic idea of utilitarianism is agreeable.)
The reason that act utilitarianism is better than rule utilitarianism is that rule utilitarianism falls into the same trap that original act utilitarianism attempts to avoid: rules that don't actually benefit people in all cases and the blind following of rules in specific cases where they wouldn't actually be beneficial. In a way, rule utilitarianism is watered down in such a way that I'd hesitate to actually call it utilitarianism. All actual laws are pragmatic to at least some extent. Therefore, it seems that the only thing utilitarian about rule utilitarianism is the degree to which pragmatism factors into the ostensible justifications for the actual rules. But in the end, rule utilitarianism will lead to instances of the very problems that the original act utilitarianism originally sought to avoid, because the general rules will not be able to maximize happiness in all cases.
Note, I'm not a utilitarian. After certain untouchable moral rules are set (e.g., no murder, no theft, etc), I'd say that rule utilitarianism is great for filling in the rest of the more mundane stuff. There seems to be a history of "utilitarians" trying to come up with modified versions of utilitarianism so as to make it more agreeable to their own intuitions or to the intuitions of the greater population. I'd suggest that those people should actually drop the label utilitarian and just incorporate utilitarian elements into their chosen moral system as they see fit. E.g., the copyright system, as it was originally conceived, seems to be fairly utilitarian in structure, but that doesn't mean the entire justice system of the United States is or has to be utilitarian (Sandra Day O'Connor has a quote, the full text of which is at the end of this write-up, explaining that copyright was primarily intended to benefit the public good by spurring progress in science and the arts, not to provide a perpetual income stream for the copyright holder via a monopoly on a work and all of its derivatives).
Note also that some rule utilitarians argue that because of human nature, rules are necessary for order and happiness, and thus, long-term happiness is maximized by rule utilitarianism even if it dictates certain acts that don't provide the highest immediate return of happiness. Fair enough, but this is still essentially an act utilitarian argument, just with a longer than usual time frame. There's also the argument that society is made happy by doing things that seem to be intuitively moral. Again, fair enough, but this is still an argument for maximizing happiness, not an argument fundamentally against the idea that maximizing happiness should be the greatest moral imperative.
The actual text of the above mentioned quote:
The primary objective of copyright is not to reward the labor of authors, but "[t]o promote the Progress of Science and useful Arts." To this end, copyright assures authors the right to their original expression, but encourages others to build freely upon the ideas and information conveyed by a work.... This result is neither unfair nor unfortunate. It is the means by which copyright advances the progress of science and art.
-Justice Sandra Day O'Connor (Feist Publications, Inc. v. Rural Telephone Service Co., 499 US 340, 349 (1991))
(The quote was pulled from the URL http://www.ntf.flinders.edu.au/TEXT/copyright.html. Thanks Chase. And thanks to rootbeer277 for catching a couple of typos.)