<p>I have many years of experience in Java including Swing, Servlet and JDBC, but have never programmed for a Java EE server.</p>

<p>Many job advertisements from large companies are specifically asking for Java EE experience.  Are there specific skills or development environments that I should learn to qualify for these kinds of jobs?</p>

Those jobs are most likely asking that you have experience with one or more of the major app server vendors, i.e. WebLogic, WebSphere, or JBoss. It's a bit different from what you're used to with servlets & JDBC, but it's still just Java. Typically you have to deal with things like JNDI lookups, dependency injection, messaging queues, maintaining application state between transactions, object/relational mapping, etc.
I have found that for most Java EE projects, it's more about knowing how components and systems are configured and how they work together than it is about any specific programming skill. And you also spend a lot more time waiting for the server to start up ;);