The short answer: No. Rainshadow deserts are found on the leeward (downwind) side of mountain ranges, not the windward side. The windward side is typically the wettest place in the region.
When moisture-laden winds hit a mountain range, they are forced upward. As the air rises, it cools adiabatically, condensing into clouds and releasing precipitation—rain or snow—on the windward slopes. This process, called orographic lift, creates lush, wet environments (think temperate rainforests like the Olympic Peninsula or the western slopes of the Andes).
Once the air crosses the summit and descends the leeward side, it compresses and warms. This warming dramatically lowers its relative humidity, evaporating clouds and suppressing precipitation. The result is a dry "shadow" cast by the mountains—often a desert or semi-arid steppe.
So, in middle latitudes (and all latitudes), the windward side gets the rain; the leeward side gets the desert.