Peel cucumber and slice very thin. Place cucumber slices between two paper towels set in a colander. Allow liquid to drain, about 10 minutes.
In a small bowl, blend together cream cheese and mayonnaise. Add next four ingredients and blend well. (You can do this by hand or with an electric hand mixer.)
Spread approximately one tablespoon cream cheese mixture on each slice of bread. Arrange cucumber slices generously on one slice of bread, then top with a second slice of bread. Repeat until you have five sandwiches.
Stack three to four sandwiches together and gently slice off the crusts with a serrated knife. (I like to do this with an electric knife. It makes it easy to cut gently.)
While sandwiches are stacked, gently slice them on a diagonal for half sandwiches, or cut across again to make quarter sandwiches.