Sorry for you, there is no equivalent in JSR-310 to JodaTime-Interval-class. I have doubts if this will ever come, but project lead Stephen Colebourne considers at least to support it in the scope of his external library Threeten-Extra, see this issue.
If you are happy with JodaTime you should keep it. Not everything in JodaTime is ported to Java 8 (Interval is not the only issue).
Update from 2014-12-13:
The situation in Java-8 has not changed, but you might also consider other external libraries beyond Joda-Time. Either Threeten-Extra which now includes a very simple interval class since v0.9 (see the other answer of S. Colebourne here) or my library Time4J which offers the range package since v2.0.