Whisk together the lemon juice, olive oil, garlic, oregano, mint, cumin, paprika, salt and pepper in a medium bowl. Dice the chicken and remove any excess fat or other tough membrane etc. Try to make the pieces relatively even in size. Add the pieces of chicken to the marinade and mix so all the pieces are well covered. Cover and put in the fridge for at least 2 hours, but a bit longer is also good up to overnight.
Before you start cooking, cut the chunk of cucumber in half, remove the seeds in the middle, if they are large, and discard. Coarsely grate the cucumber then gently squeeze most of the liquid. Put the drained cucumber in a small bowl and add the yogurt, lemon juice, olive oil, salt, garlic and mint (or dill). Mix together and set aside until needed (chill if any more than around 15-20min).
Remove the chicken from the fridge around 20 minutes before you plan to cook to let the chicken come nearer to room temperature. If using wooden skewers on a flame grill/BBQ, soak the skewers. Thread the chicken pieces onto skewers, not pushing the chunks together too tightly. Warm the grill to a medium heat then place the skewers on the grill (you can also cook them on a grill/griddle pan). Turn once or twice and cook until cooked through, around 3 - 4 minutes a side depending on size of pieces of chicken.
Once cooked either make a pita wrap by removing the pieces from the skewer and placing on pita with a couple slices of tomato and red onion and a good spoonful of tzatziki. Alternatively serve the skewers with pita and tzatziki on the side and serve eg with salad and grilled vegetables.