When I was a kid growing up in the 50s and 60s, there were a lot of western series on TV but not anymore. What was the last western series on any of the Big Three U.S. networks?

What do you define as a western?

https://en.wikipedia.org/wiki/Westerns_on_television

The last western from the 50s and 60s was "Gunsmoke" which finished its run in 1975. Ironically it was one of the first of that batch also and with 20 seasons was one of the longest-running scripted primetime programs in US history.

"Dr. Quinn, Medicine Woman" (1993-1998) was a more recent show set in the US west of the late 19th century (the "Wild West" era).

The last true Western that I can think of that aired on one of the Big Three networks in the US was "The Magnificent Seven", which aired on CBS for two seasons, ending in July 2000.
https://en.wikipedia.org/wiki/The_Magnificent_Seven_(TV_series)

Now, there have been Westerns since then that aired on cable networks, most notably "Deadwood", which aired for three seasons on HBO (2004-06, plus a movie wrap-up on 2019) and "Hell on Wheels", which aired for five seasons on AMC (2011-16), as well as Westerns on streaming services such as "Quick Draw" on Hulu (2013-14) and "Godless" on Netflix (2017). But the question asked for the last Western on the Big Three, so these do not count.
https://en.wikipedia.org/wiki/List_of_westerns_on_television

Dr. Quinn Medicine Woman (1993-2008) and Walker Texas Ranger (1993-2001) were the last two westerns on ABC, NBC, or CBS.

"Dr. Quinn, Medicine Woman" was indeed a Western on CBS, but it wrapped in May 1998, two years earlier than "The Magnificent Seven". There were two one-shot TV movies that followed it, in 1999 and 2001, but the series was completed before them.
https://en.wikipedia.org/wiki/Dr._Quinn,_Medicine_Woman

"Walker, Texas Ranger" is a police drama set in the present -- which is why it's been revived in 2021 as "Walker" on The CW (which is a free-TV affiliate of CBS), despite Westerns generally being dead as a doornail. Although it's set in Texas, it's not a Western.
https://en.wikipedia.org/wiki/Walker,_Texas_Ranger
https://en.wikipedia.org/wiki/Walker_(TV_series)

It's meaningless to restrict the selection to the 'big three' today. Most people would agree that the best shows aren't on the big three anymore. It's also pointless to quibble about what qualifies as a 'western.' 'Yellowstone' is a modern western that is still ongoing. Quality trumps quantity every time.

Western:

2 often capitalized : a novel, story, motion picture, or broadcast dealing with life in the western U.S. especially during the latter half of the 19th century

https://www.merriam-webster.com/dictionary/western

The Young Riders was about the Pony Express. It was on from 1989-1992. It was a "real western" in my opinion and on ABC

https://en.wikipedia.org/wiki/The_Young_Riders