Soon-to-be attorney.
You most likely cannot copyright a cubing technique (at least not in the United States). Sec. 102(b) of the Copyright Act states that copyright protection does not extend to a process, system, method of operation, concept or principle. It is highly likely that a cubing technique will fall into one of these categories.
You can most likely copyright a book describing the technique, but the copyright protection will only extend to the expression of your idea in the book and not to the method itself. See
Baker v. Selden, which is an informative case. If this happens, people will be prohibited from copying your book directly. But, they will be able to use the method you mention in the book as they please.
Patent law is most likely not going to help you either. To get a patent, you have to satisfy many statutory requirements (statutory subject matter, utility, novelty and nonobviousness). A cubing technique will most likely fail every one of these requirements except utility.
Further, prosecuting a patent is usually a time-consuming and expensive endeavor. Patent prosecution has to make economic sense. You don’t automatically get money when you secure a patent. The money comes from individuals or entities that pay to purchase or license your product. You can also get money if you sue for infringement. Assuming that your patent is granted by the Patent Office, you’ll make money off of charging people to use your method. If they don’t pay up, you’ll sue them for infringement in hopes of recovering. Nobody serious in our community would ever consider charging others or suing fellow cubers for turning the faces of a puzzle in a particular fashion. As a result, you'll end up paying a great deal to get a patent but not derive any economic benefit.
In summary, you’re probably not going to get much protection for your ideas unless you keep them a secret. However, secrecy is generally discouraged in our community. Also, most people in our community lack the expertise and know-how to fully develop polished, novel, usable methods. We exchange ideas in hopes that other people (like Kir) can help out or shoot down the method in a beautifully offensive but truthful manner.
Wide dissemination as early as possible is preferred here. But, you are free to do whatever you please. I think seeking IP protection is out of the question, though.