This is a very simple recipe, so you can play with it. I use porcini mushrooms here, but any thick mushroom works. See the headnotes for options. Once made, these will keep a couple days in the fridge.
Get your grill hot, and make sure the grill grates are clean.
Make sure the mushrooms are clean by wiping them with a damp paper towel and/or trimming dirty parts with a knife. You are looking for largish pieces that can stand up to the grill, so anything about 1/4 inch to 1/2 inch is good. I mostly prefer thick slices of porcini, or portobello caps.
Toss the mushrooms in oil and salt them well. Lay them cut side down on the grill and let them sear until you get good grill marks, typically about 3 to 5 minutes. Turn them over and give the mushrooms another 2 to 3 minutes. You want browning, but not dried out mushrooms.
Move the mushrooms to a bowl and toss with a bit more olive oil, crumbled oregano leaves, lemon zest and juice, and some black pepper.