There are direct bus services from Bodrum.
Earlier this year a friend travelled to Bursa on the Istanbul bus but I'm not sure if she had to change somewhere along the route (possibly Soke but it wasn't Izmir).
Thanks for all the comments; they've been very helpful.
It looks as is there are a couple of coach services that go between Bodrum and Bursa (on their way up to Istanbul), Kamil Koç, and Varan. It's a little unclear whether a change is required on the Varan service, but I'll do a little more research.