Well and it actually is beyond the Mirkwood. The lands east of Mirkwood are basically a steppe, where the northern parts are possibly a bit more lifeless (but that is also due to the local fauna, like... Dragons
). I guess Mirkwood is such a big piece of forest that it holds the moisture, especially taking into account the fact that the waters from the Grey Mountains likely descend there. I think Southern Mirkwood, funnily enough, might be somewhat closer to Mediterranean-type forests (minus the sea factor). Then, once again taking into account "unnatural" factor, just south of Mirkwood there used to be the Entwives' gardens, so... a "cultivated" land in the best sense of the word, so I guess originally they would have supplied their country with water by some ingenious yet "nature-friendly" means, which may have in turn reflected even on the neighbouring places, like Southern Mirkwood. Just for the record, before somebody voices it, personally I don't think the Wood-Elves would be tampering with the climate (they are not of that sort of folk to do that). But anyway, it may be also worth considering that there is Anduin and lake Rhun and if there were winds coming towards the mountains also from the East (we don't know exactly how the atmospheric currents worked in M-E, but the "East Wind" seems to have been important in Rhovanion, at least the Dwarves basically sing about it in Beorn's house, describing the route of some air currents
), then there will be water coming down also on the western side (mostly Anduin Vales, though - that explains the Great River just next to the Mountains).
But generally, perhaps Rhovanion beyond Misty Mountains = similar to Siberia beyond Ural? And now I don't have a map at hand, so I cannot even check how would the distances from the Sea differ there, but it comes to me as one similar thing.