Step 1Place the beets in a large saucepan with plenty of water to cover. Bring to a boil and cook at a fast simmer until the beets are tender enough to piece easily with a sharp knife, about 45 minutes. Drain, rinse under cold running water, shake off excess moisture and set aside.
Step 2Trim the cucumbers and slice them a little less than one-fourth-inch thick.
Step 3In a large bowl, mix the olive oil, vinegar, garlic, herbs and salt. Add the cucumbers and stir to coat well.
Step 4Remove the cucumbers with a slotted spoon, draining any excess dressing back into the bowl. Arrange the cucumbers on a platter in a broad layer.
Step 5Using your fingers, slip the peels off the beets. They should come off easily. Slice the beets one-fourth-inch thick, add them to the remaining dressing and stir to coat well. Remove the beet slices with a slotted spoon and arrange them in an oblong mound on top of the cucumbers, centering them so there is cucumber showing around the edges.
Step 6Sprinkle the feta over the top and serve immediately.