This will be third year we're trying to get a grape cutting to grow. We're even 'cheating' by buying the cheap ($5 this year from Tractor Supply) rooted twigs from the big box stores that already show a bit of growth and have a some roots. We every get one and plant it and every year it doesn't make it. Any ideas or advice?

4 yrs back I bought Grapes at Walmart, standard retail grape plant, almost bare root, in a plastic bag. Last year I had 1 small cluster on 1 vine.  This year I have multiple clusters on all vines, including clusters on 1 air layer.  Grapes, and most other fruits are not as quick to produce as figs with some trees such as Pear or Apple requiring 4-5 yrs.

You also should consider the weather/zone/seasons in your expectations.

Patience is needed when dealing with many things, especially when dealing with plants.

I have the best luck with grape cuttings by planting directly in the ground. 
Prepare the area get rid of any weeds.  Cut the bud off on the bottom of the cutting so you only get 1 or 2 buds from the top to grow. 
Water often and 70% success. 
The challenge is keeping the weeds out so they can get nutrition and grow strong.

I'm not talking production. The plant/twig just dies... :(

Do you plant the grape in the same location every year? In the ground or in a pot? If so, how much/little sun does the plant get? Do you fertilize?

Bunchgrapes.com

Pino, do you start the cuttings in sun, partial sun, or shade? Do you cover them with a jar or anything while they are rooting? When are you starting them?

You might look for a local nursery that sells what works in your area.(soils vary)

I know that the lambrusca (Fox grape)and vinifera crosses like catwaba, concord etc. Work well in NYS...but your care and conditions can vary...

You may need some one from your region to chime in...or better yet ditch the cookie cutter stores and start calling the locals.