I could take or leave the smash hit “Cars,” but I loved the rest of Gary Numan’s ***The Pleasure Principle ***album. That one’s well worth a listen.
Split Enz has been mentioned more than once; I recommend Time And Tide.
Also check out Jane Siberry. She’s a Canadian artist, not too well known in the States. Try No Borders Here. If you like The Flat Earth, I think you’ll like this quite a bit.
Give The Age of Plastic by the Buggles a go. It really is a well-produced and evocative album, IMHO.
Talking Heads or Tom Tom Club
Kate Bush
Already love Loreena McKennitt, Talking Heads, and Tom Tom Club (Byrne somehow found one of the best batteries in the business). “I Scare Myself” on Dolby’s Flat Earth sent me back to Dan Hicks and I dusted off Last Train to Hicksville for the first time in about 25 years; till remember most of the lyrics.
Did nobody mention Brian Eno?
I always liked The Sugarcubes. (The band Bjork was in before her solo career.)
How about recordings younger than my children, born in '89? Sigur Ros yes, Bjork yes, but Sugarcubes no.
LCD Soundsystem - electro dance, clever lyrics.
Spoon - more guitarry than synthy, but amazingly consistent high quality output.
Darwin Deez - Clever, catchy songs - Radar Detector being my favourite.
St Vincent - new album is hotly anticipated, the album she did with David Byrne is very good too.
MiM
I don’t know music very well, but I’d say Fatboy Slim, and Moby.
(Here is the Moby song)*
*Not actually by Moby
This Mortal Coil - mostly electronica/dream pop covers of the likes of Tim Buckley and Alex Chilton, or original instrumentals.
For goofy pop there’s always Men Without Hats. “Rhythm of Youth” was a decent album for its time; if you liked “The Safety Dance” you’d probably also like “Living in China” or “Ideas for Walls” which are all much of a muchness.
Also, Yaz (or Yazoo, depending on which country you’re in) had a phenomenal album in Upstairs at Eric’s. A bit gloomier than Dolby but I loves me some Alison Moyet.
I read that as ‘Men at Work’. Which is another good choice. Especially since their two albums are slickly produced and hold up well (IMHO). Another 80’s album that holds its value is Frankie Goes to Hollywood’s album, Welcome to the Pleasuredome
Human League is a band that hasn’t been mentioned yet but might fit the bill. Also Ultravox.
I just wanted to say that I’m touched I inspired some interest in Thomas Dolby, as I love his music dearly.
Give Depeche Mode a try. I would start off with Violator. It is a solid album start to finish, and as my friend Bryan once said “the album could easily be about a romance, or drug addiction, but incredibly it’s both.”
Very well produced, also.
Another crisp favorite of mine would be Duran Duran - Rio
Jesus fuck, you people are stuck in the 80s! 
I see your 80s and raise you 70s Krautrock: Can! Amon Düül II! The synth paradise of Kraftwerk! English lyrics by people who don’t speak English! 
I remembered this question from last week and it came to mind while listening to another album since I thought “This almost seems like Thomas Dolby”.
It was David Bowie’s Black Tie White Noise.
One of his other 90s albums, Earthling, also has some tracks that sound like they could be Thomas Dolby songs. Little Wonder and Dead Man Walking.