I'd wondered the same thing, so here's what I found on the internet:
Juglone [in black walnut trees] is found in leaves, stems, nut hulls, inner bark and roots. It can affect plants growing under the tree or within the sphere of its root system. Remember that a treeÂs feeder roots - those fine, hair-like roots responsible for absorbing water and nutrients - can extend out two or three times the circumference of the spread of a treeÂs branches (drip line). Although plants growing immediately under the tree are most at risk, sensitive plants growing in the area of its extended root system can also be damaged or killed.

Funny your story sound familiar today of mine, I was talking to one my Neighbors today while out doing my own gardening and they are ripping up there backyard for a Garden Bed right beside a black walnut a tree, I told them and then they asked the other niegbours about it and obviously I could hear everyone was really getting pissed off about this tree in their backyards.

I got a chance to talk to one of the other neighbors that has a garden Bed near the tree.

He said he has grown Tamotos about 12 feet from the dripline of the walnut Tree with no problem.

So least make sure your not directly under the tree. Myself I'm using Rubbermaid containers. I dont have alot of space. the Dripline pretty much consumes my whole backyard.
But dont be broken up about it if your tomatoes die during half the season the Black walnut may still get em, they release the toxin threw their roots and when it rains the toxin will fall from the dripline of the tree. Hence why you dont want to be under the dripline.

Some plants though like Corn won't be bothered at all. You can a list of the those plants if you search on Google or whatever.

Roots do not go out from a tree evenly in all directions so if you are a decent distance away from a walnut tree about all you can do is give it a try and see what happens.

Could you try raised bed gardening? Would digging a trench around where you want your garden killing the roots in your yard help? Of course these are only if you are not under the drip line. Sorry if these would not work, as I am not very familiar with dealing with the Black Walnut trees.

I grow my tomatoes about 20' from the dripline of a Black Walnut and have had no problem whatsoever. You should be fine. Don't compost the walnut leaves as they may taint your blend. Best of luck.

Thanks everyone! I didn't realize I had responses, because I assumed I'd be emailed them. :)

The three I planted in the ground aren't growing quite as well as the ones in the pots, but I think they get a bit less sun, and I think I was under watering them a bit. We had three straight days of rain, and they are finally starting to grow.

We do have peonies and rhubarb in our yard, and Black Walnut is supposed to kill those dead as well. So, I'm keeping my fingers crossed.

I know this is like 9 years ago...but I have this same problem, (I also live in 5a) ((I hope this link works))

https://dengarden.com/gardening/Toxicity-of-Black-Walnut-trees-in-Ohio