We present an approach for describing tests using non-deterministic test generation programs. To write such programs, we introduce UDITA, a Java-based language with non-deterministic choice operators and an interface for generating linked structures. We also describe new algorithms that generate concrete tests by efficiently exploring the space of all executions of non-deterministic UDITA programs. We implemented our approach and incorporated it into the official, publicly available repository of Java PathFinder (JPF), a popular tool for verifying Java programs. We evaluate our technique by generating tests for data structures, refactoring engines, and JPF itself. Our experiments show that test generation using UDITA is faster and leads to test descriptions that are easier to write than in previous frameworks. Moreover, the novel execution mechanism of UDITA is essential for making test generation feasible. Using UDITA, we have discovered a number of bugs in Eclipse, NetBeans, Sun javac, and JPF.