In a large bowl, whisk together the yogurt, grated garlic, oregano, olive oil, salt, and pepper. Add the chicken and turn it until every piece is well coated. Cover and refrigerate for at least 30 minutes, or up to 8 hours for deeper flavor.
While the chicken marinates, smash the cucumbers with the side of a knife and tear them into bite-sized pieces. Chop the tomatoes and toss both into a colander with a good pinch of salt. Let them sit and drain for about 15 minutes so the salad doesn't turn watery.
Heat your grill or grill pan to medium-high and lightly oil the grates. Shake off excess marinade from the chicken and grill for 5 to 7 minutes per side, until it's nicely charred and reaches 165°F internally. Let it rest for 5 minutes before slicing.
While the chicken rests, mash the feta into any remaining yogurt marinade you set aside earlier, using a fork until it forms a chunky, spreadable paste. Squeeze in fresh lemon juice and season with a bit more salt and pepper if needed.
Shake off any extra liquid from the drained cucumbers and tomatoes, then toss them into the feta yogurt dressing along with the olives. Slice the rested chicken and plate it alongside or on top of the salad. Serve right away while everything is fresh and cool.