Best way I've seen (haven't really looked recently though) involved uploading a backup of your iPhone data to a website which extracts the messages. I'm sure a quick Google search will help locate a service like that if no one else has a better solution.
PhoneView works pretty well to see just about everything on your phone. You can copy messages and I think other things to your computer. I actually used it to cut about 1 GB off of the "other" section on my phone, because PhoneView found some videos lurking somewhere after I had deleted them months ago. I think there's a free trial to see if you like it.