A couple of autumns back, I frequently went wild mushroom forraging, initially with my Basque father-in-law, who I think had been mushrooming pretty much all his life. He explained and showed how mushrooms would appear with the full moon. They would be plentiful and fresh for a few days, then gradually diminish. By the following waxing moon, any mushrooms found would be old and/or weak. There would be virtually none to find until the next full moon, when they'd suddenly be everywhere again. Is there a scientific explanation for this?
Chris, I too would go mushroom picking with my father in Sweden, and heard the same claim from him.
That said, this is more than likely a case of confirmation bias. Numerous questions have been asked here about the moon and its effect on things (I'll get to those afterwards). The biggest argument against this is would be to look at the biology of mushrooms. Since they are fungi, mushrooms do not need light for their lifecycle, since they don't contain chlorophyll.
Fungi are distinct from plants because they do not possess chlorophyll, the green pigment that allows plants to manufacture sugar from the sun's energy; they need to absorb their food from the environment in which they live.
As a matter of fact, there is a listing of mushroom species for cultivation in this PDF slideshow. In many cases, they explicitly state that no light is needed at all. Although, they may take phototropic cues to help in directionality of growth.
Now, the thing that will allow confirmation bias to creep into when you see mushrooms is that they have very rapid and cyclical life cycles. This paper for a specific white mushroom shows a 50-55 day cycle. Other mushrooms, such as the ones listed in the cultivation slideshow, show numerous time frames from 10 to 60 days.
In reading more about mushrooms, the key takeaway that I found in numerous pages is that biologists report (For instance, University of Wisconsin):
Very specific conditions of nutrition, humidity, carbon dioxide levels and temperature must be met for primordia to form.
Image from University of Northern Illinois Biology Department
Another trend in confirmation bias is that people may pick specific times to pick mushrooms based on those conditions of nutrition, humidity, carbon dioxide levels and temperature, such as after a rainy period, during specific months, etc. Add to those time phased events the natural human condition of being very susceptible to any type of confirmation bias, and there you have the claim.
As I said, there are oodles, and oodles of things people try to tie to moon phases, and in each case, there seems to be no actual correlation, only a perceived one.
In other words, there really doesn't seem to be a lot associated with a new or full moon.