Add to this list:
X Minus One
The Twilight Zone
Alfred Hitchcock Presents
Night Gallery
Twin Peaks
The X Files
Nightmare
One Step Beyond
Do podcasts count as radio? If so:
Welcome to Nightvale
Pseudopod
Sure. Why not? 
The Prisoner
The Maxx
The Head
Robot Chicken
Lancelot Link, Secret Chimp
For radio: I always thought of Dr. Demento (and his show) as being plenty weird.
Outer Limits (2 incarnations)
Kolchak: The Night Stalker (2 incarnations)
Thriller
Twilight Zone has 2 incarnations
Amazing Stories
Max Headroom (?)
Inner Sanctum (radio)
Tales From The Darkside
Dark Shadows (2 incarnations)
Dark Room
Tales From The Crypt
Bob and Ray
Ghost Story (a Night Gallery clone)
Circle of Fear (ditto)
Mad Movies (late 80s show on Nickelodeon)
Eerie, Indiana (the center of weirdness in the universe)
Picket Fences
Somebody help me out, here:
Did/did not Anthony Quayle host a weird series in the '60s/'70s?
Have you tried his IMDb page? I’m not familiar with his hosting something.
Dr Demento was/is pretty weird.
The Soupy Sales Show was so weird, I’m amazed it was aired in the after-school hours!
Rocky and Bullwinkle pretty much set the standard of weirdness for me. I was sixteen before I understood most of the jokes!
The old (syndicated) Steve Allen Show ranged between weird to bizarre.
And speaking of Bizarre…
***SCTV*** was wonderfully weird when Harold Ramis was on it:Are you referring to Strange Report?* It was a kind of early British version of CSI, as I recall:
*Or are you just dickin' with us here? :dubious:No, no, no, I would sooner have my eyes gouged out than mess around with youse guys. I thought it was an anthology series, thought it was British, and I did not recognize anything on the IMdB page. I shall return.
How about Shatner and Nimoy’s series?
(I mean, of course, Weird, or What? and In Search Of, respectively.)
CBS Radio Mystery Theater, with E.G. Marshall, from the 70s. You can listen to these on YouTube.
Do The Avengers and Batman qualify? 
Freddie’s Nightmares
Friday the 13th (no relation to the movies)
Lights Out
Quite Please
The Sealed Book
The Shadow
Chandu the Magician
Exploring Tomorrow
Mindwebs