Marinate the chicken all day, or overnight - or freeze it already cubed, in its marinade, until you’re ready to grill. The tzatziki will get more intense after a day or two in the fridge too.
To make the souvlaki, combine the chicken, yogurt, oil, garlic, oregano and salt in a bowl or heavy zip-log bag; seal or cover and refrigerate for several hours. Soak bamboo skewers, then thread the chicken onto them, leaving room for a handle. Grill over medium-high heat until charred and cooked through.
To make the tzatziki, coarsely grate the cucumber(s) onto a double thickness of paper towel; gather it up into a bundle and squeeze out as much liquid as you can. In a medium bowl, stir the drained cucumber into the yogurt, along with the garlic, lemon juice and salt. Refrigerate until needed.
Serve the souvlaki as soon as it comes off the grill, with the tzatziki.