Dish draining closets are popular in Finland -- very popular. They are not very popular in the rest of the world -- not at all. This is a pity, because they are an excellent example of good design in the kitchen. The cabinets mount directly over the kitchen sink, and instead of shelves they have racks for drying dishes. As you might expect, the cabinet has no bottom so dripping water falls directly into the sink. Other than these modifications, it looks just like any other kitchen cabinet. Some versions have only the lower few shelves converted to drainboards, with the upper shelves constructed in the regular fashion.
This has a number of benefits -- you don't waste counter space with a dish drainer, and if desired commonly used dishes can be stored in the dish draining cabinet, saving a step in cleanup. It also holds about three times as much as a standard counter-top dish drainer. The cabinet is particularly useful in Finland because, for reasons unclear, the dish drainer is traditionally placed on a table, so that it cannot drain into the sink, and hand drying is common.
They were invented by the famous Finnish inventor and work efficiency expert Maiju Gebhard, who was heading the Finnish Association for Work Efficiency at the time. She calculated that the average housewife spent 30,000 hours of her life washing and drying dishes, and developed the drying cabinet to help reduce that time. They were first introduced in 1948, and have since become a standard fixture in kitchens across Finland.
Draining closets are slowly spreading across Europe and Asia, and are present to some extent in Poland, Italy, Spain, Germany, Israel, Singapore, and parts of Russia, among other locations. They are strangely absent from other parts of the world. I have not yet seen one in America or Canada. This is probably due in large part to the NA tradition of having a window over the kitchen sink, and the widespread use of dishwashers.
In Finland these cabinets are called astiankuivauskaappi.