Combine the fresh lemon juice, olive oil, red wine vinegar, oregano, salt, red pepper flakes, and freshly ground black pepper in a small container with a tight-fitting lid. Shake vigorously for about 30 seconds until the dressing is well emulsified and the flavors are combined. Set aside until ready to use. I prefer using premium olive oil here since it's the star ingredient—the quality really shines through in such a simple dressing.
While the dressing sits, prepare all the components for assembly. Cut the feta into 32 cubes (roughly 1/2-inch each), peel the cucumbers and slice them into 32 rounds about 3/4-inch thick, halve the kalamata olives lengthwise, and keep the cherry tomatoes whole. Arrange all prepared ingredients on a clean cutting board or work surface so everything is ready for quick assembly.
Working with one toothpick at a time, thread a cherry tomato onto the pick first, then follow it with a cucumber round. Position the olive half with its flat side facing outward for visual appeal, and thread it onto the pick next. Finally, poke the toothpick halfway into a feta cube so it sits securely on top. Repeat with the remaining ingredients until all 32 bites are assembled. I find it helpful to work in assembly-line fashion—insert all tomatoes first, then all cucumbers, then all olives, then all cheese—rather than completing one pick at a time.
Arrange the assembled bites on a serving platter. Either drizzle the dressing lightly over the bites just before serving, or transfer it to a small dipping bowl on the side so guests can control how much they use. Serve immediately for the best texture and flavor. The bites are best enjoyed fresh, while the vegetables are still crisp and the feta is cool.