Preheat a gas or charcoal grill to 375°F; if using coals, wait until they turn white-hot.
Meanwhile, combine the crabmeat, Greek yogurt, lemon juice, oregano, and some salt and pepper to taste in a mixing bowl; stir to combine. Cover and chill until ready to serve.
In another mixing bowl, combine the mushrooms, oil, and plenty of salt and pepper to taste; toss to combine.
When the grill is ready, brush the grates with some oil. Spread out the mushrooms on the grill and cook, turning once, until lightly charred, 3-4 minutes.
When ready, remove the mushrooms from the grill and stuff with the crab tzatziki; serve right away.
