Are there more mushrooms during full moon?

Larian's answer suitably covers some logical issues with the claim, which is enough to perhaps comfort you in your skepticism, but it doesn't empirically answer the question for us.

It seems the association of the moon with mushrooms is ancient. Some ancient knowledge is derived from observation, but most of it is steeped in superstition and religious/occult dogma. The lunar/mushroom connection is no exception. A shallow delve into this topic quickly leads to pages discussing astrological signs, "mysterious" auras, and "ancient wisdom".

In terms of science, however, mushroom growth does not appear to be affected by the moon. In a study of 1715 mushroom harvest records, researchers in Switzerland found no relationship between harvest yields and lunar cycles.

Here we examine a total of 1715 dated mycological records, collected between 1990 and 2007 in five long-term observation plots in Switzerland to test for a possible relationship between lunar periodicity and mushroom yields. No such relationship was found, which means the claim that the moon phase influences mushroom production is based on myth.
Egli, S., F. Ayer, and M. Merlini. "More mushrooms under a full moon–myth or reality?." Sydowia 63.1 (2011): 23-33. - ResearchGate, abstract only
Full PDF Link

What mushrooms do require for successful growth is moisture, position (with regards to "food" mostly, as fungi are heterotrophic), soil integrity, and temperature, all of which may be more or less important depending on species. Light, from either the moon or the sun, does possibly contribute to growth direction, like virtually all plants, but does not affect growth robustness, as any simple "dark" growing experiment many of us have done as children shows.

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.

Are more crimes committed during a full moon?

Do people with mental illnesses feel the effect more during the full moon?

Are more babies born during full moons?

Is the human menstrual cycle related to the lunar cycle?

In other words, there really doesn't seem to be a lot associated with a new or full moon.