"Spa Water' or just a big jug of cold water with cucumber slices in it is a wonderful thing. I only do it with unwaxed cucumbers and feel the peel is a big part of the taste.
There are alcoholic drinks like a Pimm's Cup Cocktail that are garnished with a cucumber wedge. Makes a nice Bloody Mary stirrer too. What can I say -- my younger self may have been vegetarian but drank like a fish!
With the many immigrant people coming to Seattle, I was introduced to the idea of it being cooked. Now when I make something like Sweet and Sour Pork, I include a cucumber with the other vegetables. This is a good use if it is a bit seedy, simply peel and scoop out the seeds and then cut up and add to the mix, along with the peppers. Typically the pieces are reasonably large, like 1" cubes. When you first encounter them you know they are there but aren't quite sure what they are...