Tenny (or tennie) runners are tennis shoes. The term is said to be used in Texas, but is also popular in many other states.
Dallas Slanguage
Tenny Runners
A pair of athletic shoes
Google Groups: rec.food.cooking
From: Mark J. Cintala
Date: Sun, Jul 21 1996 12:00 am
Geez. This *is* a neat thread.
I’m from PA, now living in TX. I know you wonderful RFC folks will find it in your hearts to forgive me if I’m kind of immune to some stuff involving pronunciation. I won’t get started on the Texas dialect, but I will deposit these as I go:
UMbrella, INsurance, tenny runners (if you don’t know what they are, I can’t say I blame you—I was agog when I found out, myself).
How to Talk American
by Jim Crotty
Boston: Houghton Mifflin Company
1997
Pg. 342:
tenny runners: any pair of athletic shoes. What are called “tennises” in northern Florida.
18 July 1965, Tri-City Herald (Pasco, Kennewick, Richland, WA), pg. 18 ad:
Tennie Runners.
8 September 1967, Modesto (CA) Bee, pg. A7:
The women, looking somewhat more professional, wore their tenny runners, black slacks and blouses, and pink vests—bearing the name of their sponsor, the Edd Tide Lounge.
12 December 1971, Tri-City Herald (Pasco, Kennewick, Richland, WA), pg. 7:
Or, every man in his tastes, opined the little old lady from Boistfort, who fancies tenny runners, as she kissed the nearby Pe Ell bovine by name Bessy.