I am swimming against the tide but I quite like checked exceptions - when done properly. I find them self-documenting and they force engineers to consider the possibility that their request cannot be serviced.
My partner, who is a lawyer but a wannabe geek, laughs when I tell her that lots of the problems I encounter at work are because somebody did not consider a certain set of circumstances. She tells me that in law, it is a very poor contract that does not cover all eventualities.
When I discuss my unpopular view with other engineers, I often hear that if you can't do anything then you have no choice but to re-throw an Exception. This is true but people often underestimate the number of options still open to them upon catching one.
For example, in the event of a database deadlock, one of the threads will immediately be rolled back. The JDBC driver will throw an SQLException but then that thread can try again.
To be fair, Java's JDBC API has left something to be desired. For most JDBC drivers, an update that violates a constraint and is never going to succeed throws the same exception as one that was deadlocked and might work upon a retry. (A class was added to the SQLException hierarchy to address this - java.sql.SQLRecoverableException - but it only appeared as late as JDK 1.6).
I am content to adopt the house-style. If my client prefers RuntimeExceptions then so be it. But an area where I feel checked exceptions should be mandatory is when one team creates an API for another.
A problem I saw recently was when the core team wrote a Java API and published it via Java RMI. A second team called that API but upon encountering a non-deterministic deadlock was surprised when the JVM barfed with an Error. It transpired that DeadlockLoserDataAccessException was not in their classpath - a Spring class that was thrown when Spring converted the SQLException to something a little more descriptive.
With checked exceptions, APIs are not only somewhat self-documenting but better define what the client needs in its classpath.
This proposal is not a panacea - objects passed over the wire might reference non-standard implementations. For instance, objects re-hydrated with Hibernate might have Hibernate implementations of standard Java collection interfaces (you might want to look at Gilead to clean up the object graph in this case).
[Interestingly, the proposal to use Gilead on this particular project also prompted a discussion on what was reasonable tech debt. Not all debt is bad - a mortgage allowed me to buy my house. Only when it becomes unmanageable is it a problem. When the team who wrote the API that passed Hibernate re-hydrated objects across the wire first started coding, they were the only clients of it. Therefore, they considered the use of Gilead as unnecessary. "Do the simplest thing that works," they said and put Hibernate in their client's classpath. The problem became a little more intractable when a second team used the API since they used a different version of Hibernate.]
One last point: if you publish APIs and both you and your clients use Maven, you can have the pom.xml of the API module to pull in any necessary third-party libraries as transitive dependencies. However, do remember that not all teams use Maven. What's more, not all APIs use just Java.