while i'm now living in san jose, i grew up on the big island from '79 - '91. i rode only street back then, a few roads might be especially fun to try on a klr;
chain of craters - it's paved but, packed with smooth sweepers.
mauna loa strip - national park side, watch for on-coming traffic, bumpy pavement & tight corners.
mauna loa observatory road - must be real dual sport stuff, we bottomed out a nissan sentra rental in deep gravel near the beginning and came to our senses, i've never been...
cane haul / kealaiki road between pahala & naalehu - wanna get away? broken pavement and nice views of one of the least developed parts of the island
mauna kea summit - could a klr handle 13k+ feet? would the telescope protestors at the base allow passage? if yes to both, other worldly views if the clouds part!
waipio valley - if you and your brakes survive the descent into, a trip back into old hawaii with lots of stream crossings, be aware of the private property.
caveat - i only rode a few of these with my '87 vfr while i lived there. there's a reason you see a lot of 4wd trucks despite the gas prices.
i'm visiting the folks next month and would love to hear some of your picks!