Oh, and I almost forgot. Along the lines of Shonen Knife, Dressy Bessy has also written a song for the Powerpuff Girls.
Frente!'s first album, from back when the “!” was still part of their name.
The Cardigan’s first album, the only good one they ever made, sadly.
Danielson Famile are pretty cute, though they’re kinda spooky, too.
Mates of State are very cute.
But something makes me feel like I’m off the mark here somehow.
Nearly a whole page of replies and no one’s mentioned The Pooh Sticks yet? Million Seller is about as cute as it gets.
Though Shonen Knife was the first thing that came to mind for me as well. And my nominee for cutest Jonathan Richman song is still “Not Yet Three”.
If you want some nice indie music you should check out Easyworld. They’re a British band i’ve been into for about a year and they have some really nice happy songs. I recommend Try not to think , Junkies and Whores (if you want something thats just gorgeous), Bleach, You and Me, Everyone Knows and Better Ways to Self Destruct.
Shonen Knife is great…my favorite song of theirs is “Cookie Day”…“cookie day, cookie day, it’s a beautiful cookie day…” 
Also, I would go see Jonathan Richman in concert if all his tour dates weren’t in Italy. rrrr.
Sixpence None the Richer
Bel Canto
The Rembrandts
early Cindy Lauper
OK, I have to ask… I have a Powerpuff Girls song that’s tagged as Apples in Stereo. It’s calle “Signal in the Sky (Lets Go!)” Is this by the Apples, or is it mistagged and actually by another band?
There’s a whole recordful of Powerpuff Girls-themed songs by different artists. The Apples in Stereo song, the Shonen Knife song and the Dressy Bessy song are all included. There’s also a Frank Black song and many others.
Lemme just chime in with a couple addendums…
The Vaselines - These guys fit your description, and IMHO, are required for the completion of any decent record collection. They become somewhat popular in their own right, but their fame was accentuated by Kurt Cobain’s heavy praise for them, and the inclusion of two Vaselines covers on Incesticide (“Molly’s Lips” and “Son of a Gun.”)
All Girl Summer Fun Band - Name says it all. Their web site has several mp3s to check out.
Also, it’s worth having a look at allmusic.com and searching on “twee pop” in the musical styles category. If you didn’t know already, that’s what this sort of indie bubblegum pop is generally called. It’ll refer you to a lot of cutesy artists, most of which have probably been already mentioned.
I must also recommend Shonen Knife. Their version of the Carpenters’ “Top of the World” will make you giggle, as will the entire album Let’s Knife. Also recommended and not yet mentioned is Morning Musume, a group of 13 young girls in a sort of Japanese Menudo. Extremely poppy, extremely catchy, and very good for making you smile when you feel down.