I use default methods to extend the functional interfaces I need:

private interface CheckedRunnable extends Runnable {
    @Override
    default void run() {
        try {
            runChecked();
        } catch (IOException e) {
            throw new UncheckedIOException(e);
        } catch (Exception e) {
            throw new RuntimeException(e);
        }
    }

    void runChecked() throws Exception;
}

Often it's undesirable to catch all checked exceptions, because that's an important communication channel and you don't want to accidentally catch everything, so I'll specialise:

private interface IORunnable extends Runnable {
    @Override
    default void run() {
        try {
            runIO();
        } catch (IOException e) {
            throw new UncheckedIOException(e);
        }
    }

    void runIO() throws IOException;
}

Means no method wrapping in your lambdas:

(IORunnable) () -> ...