Do lunar rhythms have really impact on plant growth?
Great question! The way I will do this is by explaining a bit about what moon planting is, then I will go through the articles I have found one at a time, to provide the different persepctives on moon planting, as it is called.

What is moon planting?

Moon planting is a methodology of planting, in which the lunar cycles are used as guides to know when to plant.

How might it work?

The first article is a great article about a gardener who tried to do moon planting, but it did not work for her, until she realized she did it wrong. Then, she found out the logic behind why lunae planting works. It really doesn't have to do with lunar cycles. Here is her explanation:


  Let’s look at a possible scenario. Gardener I is a moon planter; Gardener II isn’t. Both gardeners wait till spring to plant their beans. (No sensible cool climate gardener will plant beans in winter. It’s too cold for them to germinate, and many seeds will rot or be taken by ants.) But come the first warm spell Gardener II succumbs to one of the great spring urges and plants the beans at the first hint that spring has arrived. Gardener I, on the other hand, waits till the next good moon planting time before planting the seeds. Early warm spells are usually followed by another cold one… and again seed planted too early may rot. Even if it doesn’t, plants that suffer any set-back when they are young usually don’t do as well as plants that have flourished right from the start. (The set-back can be from cold, boggy soil, snail or scale attack — the effect is the same). So counterintuitively, beans that are planted later in spring will probably do better than beans planted too early. 


The opposite works too when planting in autumn. By the lunar planting schedule, it is better to grow earlier, and since the soil is warmer, the plants will grow better.

In the second article, the author reviews the research done on the effect of lunar cycles on reproduction, germination, and more. However, only the work of a researcher, L. Kolisko, shows evidence for the benefits of lunar planting. Therefore, the conclusion was lunar planting are not significant to affect crop yield.

However, in another article, it shows that there are many cycles in plants that are affected by the lunar cycles and that for some organisms, the lunar cycles do affect reproduction, germination, and production of different metabolites. Therefore, lunar planting might have an effect in that way as well.

Conclusion

The conclusion is that lunar planting may actually work, but many scientists are debating about it. So in reality, we aren't 100% percent sure about it, because there is evidence supporting it, and evidence contradicting it. 

References


http://danfingerman.com/skeptic/Moon_planting.pdf
http://www.howplantswork.com/wp-content/uploads/2009/07/beeson_1946.pdf
http://www.howplantswork.com/wp-content/uploads/2009/07/schad.pdf