[barrkel]

    interface ThrowingConsumer<T, E extends Exception> {
        void accept(T value) throws E;
    }

    // ...

    <E extends Exception> void withThing(ThrowingConsumer<Thing,E> callback) throws E { ... }

The problems are that (a) you need to add an extra type parameter for exceptions all over the place, (b) exception unification (sum type) doesn't work generically - you can say E1 | E2 in a catch block but it's not a real type, and (c) it composes poorly with existing libraries that expect Consumer<T> throughout.

For one level deep callbacks (for resource handling and the like) it works reasonably well though.

[paulddraper]

(b) is what I had in mind as the problem.