You can use slicing cucumbers instead of pickling cucumbers.
I just make sweet pickle chunks. I make zucchini relish, or at least used to when I could still have sugar. Sorry, I don't make sweet cucumber relish.
Sweet Pickle Chunks

Pour boiling water over 24 pickling cukes for 4 days. On 5th day, drain, add this syrup.Cut cukes in chunks.
8 cups sugar
4 cups cider vinegar 5% acidity
3 1/2 tsp. pickling and canning salt
2 Tbsp. pickling spices, tied in a bag

Drain and heat syrup for 3 more days. On 4th day, pack pickles into jars, seal with lid. Process in water bath canner for10 minutes.
I love these with tuna!

I was going to ask for a good sweet pickle relish recipe made from cukes, too. Something you could use in potato salad (I cheat) or on hot dogs. Something not too complicated.

I think using slicing cukes to make relish might make it too watery - unless you removed most of the seeds prior to chopping.
bcskye - the best recipe for sweet pickle relish I've seen is in the Ball Canning Book. It's outstanding! In addition to hot dogs, I use it in egg and tuna salad sandwich spreads. The chopping takes time, even when using a food processor, but the ingredients aren't very exotic and it's not complicated.

Thank you olliem, I'll go ahead and use that one. It sounds just like what I've been looking for. My cukes aren't in yet, but I got 5 from a roadside stand today for $1 and I thought I ought to be preparing for when mine do come in. I do have several picklers planted.

Pickling cukes have very small soft seeds and work very well for not only pickles, but for a relish. Regukar cukes tend to have tougher skins and biger seeds, that are not good in any pickle. Also, 'burpless' cukes are too soft for pickling uses.