Nothing. As soon as the tomato/pepper root contacts the living or recently dead root of the walnut tree it will be poisoned.

The dead walnut roots will remain poisonous for a couple of years until they rot down enough to be harmless.

You might have to use containers.

Is that just Black walnuts or the english ones?

Google says English Walnut is not as bad, but it is in second place.

http://web1.msue.msu.edu/msue/iac/greentip/blackwal.htm

You are safer the further you get away from the drip line of the tree because the roots are fewer past that distance. That area is going to be shady anyway.

I'm not under the dripline, but I had looked at several websites last night and they mentioned amending the soil to boost the plants' immunity but I didn't find much info on what specifically. I suppose I will start with adding compost and manure and work it well in there and hope for the best. Another house I lived in had a huge black walnut and I did grow toms not too far from that tree (again, until yesterday I had never heard of this problem, not sure why.....I do lots and lots of reading about gardening) Containers will not be an option for me and building raised beds just wasn't in the budget.

I'm not under the dripline, but I had looked at several websites last night and they mentioned amending the soil to boost the plants' immunity but I didn't find much info on what specifically. I suppose I will start with adding compost and manure and work it well in there and hope for the best. Another house I lived in had a huge black walnut and I did grow toms not too far from that tree (again, until yesterday I had never heard of this problem, not sure why.....I do lots and lots of reading about gardening) Containers will not be an option for me and building raised beds just wasn't in the budget.

if it is a problem, you will know as soon as it rains. The plants will wilt and/or die. The poison (juglone) is leached out when it rains.

In the future don't compost the leaves from the black walnut (that's what they say anyway)

Good luck--my last garden was not far from the dripline of black walnut. I had the toms/peppers in an 8" raised bed (no bottom barrier) and did fine.

Its rained so much here the house across the street is completely surrounded by floodwater (I like at a lake) but the veggies have been unaffected so far. At this point I'm more concerned about the over-watering from Mother Nature.

It's been a couple weeks since last posting, but so far, so good. The plants appear to have no ill effects yet and we even have baby tomatoes on some plants. I've been very diligent about removing any fallen leaves from the tree as soon as possible, and with the rain finally letting up I'm not worried about drip so much. Regardless of the outcome, I'm making arrangements to remove the tree and over the fall/winter I will be substantially adding amendments to the soil and possibly adding framework to really build up the soil like a raised bed, hopefully to keep next years plants away from the roots remaining in the ground. I wish it were easier to post pics on here, I don't want to mess with a photo hosting site.