Dean Baker has suggested to Krugman that one reason for the lower productivity in Italy is that actual employment was higher before the euro, or if you prefer disguised unemployment was rampant. Then with the entry in the euro several workers that were not legally hired were, and as a result it only seems that productivity fell a lot. That is, it looks like a lot more workers are not doing that much more in terms of production, when in fact the actual increase in employment was not that large, and, hence, the fall in productivity not as big. Certainly an interesting possibility. Below the employment in Italy and in France compared (1970=100).
Note that, in fact, up to the 1980s employment grew faster in Italy than in France, and by the mid-1990s this trend was reversed. While it is true that in the late 1990s, with the euro, it seems that employment again grew faster in Italy (which may be due to Dean's suggestion), note that the proportional increase is relatively small (about 3.9% with respect to France from 2000 to 2006), and has reversed since the crisis. It seems that most of the fall in productivity is real, and caused by the fact that Italy grows less. Again, I think these reflects overly restrictive policies and reflects the well know regularity known as Kaldor-Verdoorn Law.