This is an interesting reference, but it seems to me that the measurements are controversial enough that they are certainly not, at this point, at the stage where they force any re-think of nuclear and solar physics as currently understood.
The paper in question is
Power spectrum analyses of nuclear decay rates. D. Javorsek II et al., Astropart. Phys. 34 no. 3, 173–178 (2010), arXiv:1007.0924.
and they perform a statistical analysis on decay data from an experiment at Brookhaven National Laboratory, finding a couple of unexplained periodicities. This is part of a larger series of this sort of work by the group of E. Fischbach and J. Jenkins. Their (main) initial work on this line is apparently
Time-dependent nuclear decay parameters: New evidence for new forces? E. Fischbach et al., Space Sci. Rev. 145, 285-335 (2009).
This work is definitely controversial, and there are a bunch of papers claiming otherwise. There's definitely a bunch of papers by Fischbach et al. that claim to observe this effect, but if they're doing something wrong then they're likely to be doing it wrong throughout the series. Instead, the thing to do is to look for papers that cite the Astropart. Phys. paper above, or the Space Sci. Rev. one, and exclude self-citations. If you do this, you get a bunch of papers critical of the claims, including multiple papers claiming to directly perform similar measurements, down to one order of magnitude better sensitivity, and still come up with no unexplained temporal variation in the decay rate (example). However, you do also get several independent papers claiming to observe the effect (example).
The analysis is relatively complicated because you do expect some sort of temporal variation in your measured signal, coming not (necessarily) from the sample but from your experimental apparatus. This is because environmental variables, like the air pressure, temperature and humidity, as well as radon content, in the laboratory, have a small but nonzero effect on the detector and its electronics. The Fischbach group is aware of these things, and explicitly claims (in the paper above, among others) that they are not enough to explain the observed variations. Other groups disagree, and claim that a proper statistical analysis rules out any unexplained variation (example, example).
Regardless of whether the signal is real or not, this leaves open the question of what could be causing it - or, more pointedly, what is the proposed explanation put forward by the people that claim to observe it? Some of this could be due to new physics, but it's not completely necessary to go that far. In particular, many of the proposed explanations rely on interactions with neutrinos, either solar neutrinos (which could explain the ties to the solar cylce) or cosmic neutrinos, either 'hot' or part of the cosmic neutrino background (recently discussed on this site at Where are all the slow neutrinos?). Neutrinos do couple to beta decay, so it's not unreasonable to propose that they might be behind these fraction-of-a-percent variations - but whether they couple strongly enough is definitely a much more serious question.
If you want a good explanation for the alternative explanations proposed by the Fischbach group, the Space Sci. Rev. paper is a good place to look; for a more digestible take a good resource is these slides. In short, though, the new-physics mechanisms they propose are (i) spatial variations in the fine-structure constant, on the range of 1 A.U., and (ii) a new spin-dependent long-range force that does couple to neutrinos.