Add the chicken breasts to a bowl or zip-top back. Add the olive oil, garlic cloves, lemon, oregano, and salt. Marinate for 30 minutes up to 4 hours.
Preheat the grill or a grill pan to medium-high heat. Grill the chicken breasts for 6 to 8 minutes per side, depending on their thickness. Let the chicken rest for 5 minutes before slicing.
Meanwhile, combine the sliced cucumbers, avocado, cilantro, and green onion in a large bowl. Toss with the lime zest and juice, olive oil, garlic, and salt.
Serve the chicken, sliced, on top of the salad, sprinkle with hemp hearts.