I will probably wind up repeating most of the ones already mentioned, but some of my favorite non-Henson puppets include:
Kukla, Fran and Ollie
Captain Kangaroo
Mr. Rogers' Neighborhood (Land of Make Believe RULES! By far, this is probably one of my top favorites- mainly because I also just loved Mister Rogers so much- such a great Christian man. He's one of my Christian heroes actually. There's an incredible book about him called "The Simple Faith of Mister Rogers"- I highly recommend it.)
Howdy Doody
The Puzzle Place
Between the Lions
Lamb Chop's Play-Along
Sid & Marty Krofft
Eureeka's Castle
ALF (I LOVE ALF!)
The House at Pooh Corner (this is the early '80s puppet version of Winnie the Pooh- long before The Book of Pooh came along. Loved watching this show as a kid.)
Dumbo's Circus
The Garbage Pail Kids Movie (ah yes- 1987 classic. LOL- it's not as terribly bad as some make it out to be- it's not that great either, but it was fun to finally watch. I didn't see this one as a kid- used to watch the cartoon though.)
Howard the Duck (another infamously bad classic from 1986. It is a Marvel-comics based movie, so I'd still like to see the whole thing some day. But I really think Howard looks pretty good in the movie.)
Circle Square (a great Christian children's show I watched as a child- I loved this show! I was so thrilled when I found TBN showing episodes a while back! They've got some episodes available on their website too.)
Kids' Jamboree (another great local Christian children's show that's really hard to find any info on- I used to love watching this too.)