I've had a lot of walnut pitch build-up on my band mill. Diesel solved the problem. If you have an aux oiler on the bar tip, use diesel or a diesel/bar oil mix there and regular bar oil in your chain saw. I agree that water or soap is a bad idea for a chain saw.
I notice you have a Husq 2101. I just got a 2100 for milling wide slabs. Nice saw, if you can find enough parts to keep it running!
ASDF4240, that's right. Stickers are the spacers between the boards to allow air circulation so they dry evenly. Always sticker the wood right away, unless you want spalting (which you wouldn't want in walnut). I use 1" square stickers cut from the low-grade boards. You can rip a lot of them on a table saw in a short time.