Picnic for Kindergarten -- Recommended Games/activities for Parents and Kids?

As was mentioned before, just set up stations and let them choose freely.

I'd add:
Bubble station.
Water table.
Sand table.
Water color paint station.

Have a great, relaxed time!

At that age you probably have a limited amount of time before the kids want to just run, especially if there is playground equipment. I would have stations that kids can move freely between and not do anything that has too much structure.

Sidewalk chalk
Dip various sizes of balls in different colors of paint and throw them at a big canvas or piece of paper to make "modern art."
Face painting
Bowling (fill empty soda bottles with a little water)
Cake walk (ask each family to donate a small plate of cookies, cupcakes, etc.)
Temporary tattoo station

3 legged race
Tug a war
Red light green light
Simon says
Musical chairs
Square / line dancing
Side wall chalk murals
Water balloon fights
Balloon volleyball

For my son's end of the year kinder party last year, we had a picnic with various outdoor games. They did sack races, various types of relay races (we switched it up throughout the day, but things like three legged races, bear crawling. backwards walking, etc). You could do a ring toss or beanbag toss, put balls on a parachute, tug of war would be fun too. Anything you can play outside is fun for kids at this age. It doesn't have to be fancy at all.

I agree with set up stations but mix a race here and there. We usually try a scavenger hunt. An easy one whit stickers etc as prizes

If it is still warm, how about a water balloon toss contest?
Pitching coins into a can is cheap - if the kids get it in they get to keep the money.
EDIT - I just read Sherry's answer - perfect!!!!!! Sherry, please refresh my memory, how do you play red light/green light?

Look on the Family Fun website. Also pintrest.