There are lots of trails where you can take dogs. I've taken mine to Manoa Falls, around the Tantalus trail system, and on lots of other hikes. The thing to be careful of is the Wed & Sun pig hunting on some trails... they have hunting dogs out, and it can be dangerous for your pooch.
You might want to check this out: Dog Hiking | Hawaiian Humane Society (http://www.hawaiianhumane.org/dog_hiking.html - broken link)
Hawaiian Humane Society organizes a group hike (people w/dogs) the last Saturday of each month. At the bottom of the page, they list all the trails that they use, so you can use that as a starting list of 12 trails to check out that are dog-friendly.