Half life variations have been suspected for decades, and almost all (maybe all…but I do not pretend to have a comprehensive knowledge) have been shown to be caused by limits in experimental design. This latest set from this (now expanded) group (they did another paper on this topic a couple of years ago http://arxiv.org/PS_cache/arxiv/pdf/0808/0808.3283v1.pdf) is interesting. Any periodic variation in decay rates is of great importance regardless of whether it correlates to some astronomical phenomenon.
I guess that is a good setup for a “I’m not really going to answer your question, but” type answer.
The 2008 paper was followed by a set of measurements from the Cassini probe (http://arxiv.org/PS_cache/arxiv/pdf/0809/0809.4248v1.pdf). The distance between that probe and the sun was such that any sun effect should have been magnified. None was seen. This may disprove the solar effect, or it could say something about the isotopes used. The Cassini isotopes were different than the Purdue group’s isotopes.
The isotopes they used in the paper you cite were interesting: Mn 56: 2.6 hr, Si 32: 101 yr, Ra 226: 1,600 yr. From a quick scan of the plots it looks like the longer the half-life the greater the adherence to the annual periodicity. I also notice that the bulk of the data was accumulated before 1990, and the data after 1996 does not show the same adherence to the annual periodicity.
I’m trying not to discount these decay-rate observations, and there are good reasons to not discard them out-of-hand. Most observations of constant decay rates have been made with rapidly decaying isotopes over a short time frame. The experiments are just much easier to set up. Very long term experiments are relatively rare, and could illuminate previously unobserved phenomena. If the periodicity is on the order of a year it should only take a couple of years to confirm the data using available methodologies.