What would being a serious alternative to the scientific method imply? To be useful, it would need to make precise and observable predictions about the material world. However, if you make predictions like that, common sense implies that it CAN be observed if the predictions actually come true, and that it SHOULD be observed, at least to check whether you fooled yourself or not. But with that, you pretty much have the basic idea of the scientific method.
Sure you could question if a prediction needs to be observable and precise to be useful. If I understand Cort Ammon's answer correctly, that's what he suggests about traditional Chinese medicine. He makes the valid point that predictions concerning human beings can hardly be precise, and that's certainly a huge handicap for sociology, psychology and medicine.
But any form of medicine that claims to be useful implicitly makes the prediction that it is able to help the majority of patients. Is that a precise prediction? (With precise I mean that is possible to check objectively if a prediction came true.) Probably not, because it is also very hard to predict precisely how the illness would have progressed without interference. And if the patient feels somewhat better that might only be a result of the placebo effect.
You might be inclined to trust in a controversial theory like acupuncture (or homeopathy), but intellectual honesty would demand that you at least consider the possibility of being wrong. Even if acupuncture were all right in principle, the therapist you choose might not know what he's doing, or even be a devious fraud. Moreover, maybe only a part of a traditional form of medicine might be "the real thing", the other part might still be the result of superstition and wishful thinking. How can you distinguish between those?
If the criterium of "somehow feeling satisfied" after an intervention that promises benevolent yet inprecise consequences is not enough to evaluate a non-scientific claim, what is?
I see three possible answers. (They also apply to magical rituals and similar stuff.)
a) You blindly trust in tradition and authority. History of mankind tells us that might not always be a good idea - not in politics, and not in science.
b) You trust what feels emotionally most appealing to you. That is not exacly a promising strategy to avoid wishful thinking.
c) You apply philosophical or poetical criteria. Sadly, history has shown that this can be quite misleading. For example the ancient greek astronomers were convinced that the planets moved in circles because they considered circles to be the most perfect shapes in geometry. But modern observation has discovered that nature chose a less beautiful shape: ellipses.
The amazing thing about science is that is provides an objective way to check its own claims. You suspect that a scientist is a fool who is telling bullshit? You don't have to trust his authority, you can go and check for yourself if he is telling the truth. (Admittedly, you better not bother trying to repeat elaborate experiments like the LHC in your garage.)
In logic and mathematics claims can be checked objectively as well. But they do not provide direct knowledge about the material world. (For example, with mathematics alone you cannot decide if Euclidian or non-Euclidian geometry is the correct description of the universe, although both are "true" in a mathematical sense, i.e. logically developed out of different sets of basic assumptions.)
To conclude, I cannot see an alternative to the scientific method that has the same ability to filter out incorrect or misleading descriptions of the physical reality.