Process cream cheese and mayonnaise in a blender or food processor until smooth, stopping once to scrape down sides.
Combine cream cheese mixture, cucumber, garlic salt, and dillweed.
Spread cucumber mixture evenly onto white bread slices, and top with wheat bread. Using a 2- to 3-inch round cutter, cut sandwiches, discarding edges. Or cut crusts from bread, discarding crusts, and cut sandwiches into quarters. Store cucumber sandwiches in an airtight container for up to 1 hour before serving.