I'm hearing a series of "clunky" sounds coming from down around the bottom of the bike. Swingarmish. No variance with engine rpm, almost always from a standing start. I have not yet got down on hands and knees, but chance of swingarm wear in bearings? Chain wear into the nylon guide? I see small flecks of metal down around the swingarm pivot area...
Tim
Stop Riding it, until you can look below it. If could be the chain eating into the swing arm for various reasons. I know others will have better or more specific ideas.
I'm guessing you need to change swingarm bearings. I've been wrong before though.
I'd put the bike up on a stand or bucket and have someone hold it, and see if you can detect any play in the swingarm. I know KLRs are famous for having inadequate swingarm bearing lube from the factory, but I haven't heard of too many bearings actually going out.