Cassandra runs over a JVM, and this is all you need to get Cassandra up and running. Any platform that has the JVM can have Cassandra. At the time of writing, Java's latest version was Java SE 7. However, it is highly recommended to use Oracle Java 6 for Cassandra, to avoid unexplainable bugs due to any inconsistency in the Java version or vendor implementation.
The other thing that one should consider for the production setup is to have the Java Native Access (JNA) library. It provides access to the native platform's shared libraries. JNA can be configured to disallow swapping JVM, and hence, improving Cassandra memory usage.
The default installation of Linux systems usually contains the OpenJDK Java Runtime Environment (JRE). This should either be removed or you can keep OpenJDK, but set the default JRE as Oracle JRE. This guide will assume a 64-bit system. To check whether your system has JRE, and what version of it executes, run the java -version...