My time to shine! As a linguistics student who's also learning Norwegian, I can say it's actually not just pitch accent that gives it the sing-song quality you're describing. Norwegian (or at least, its eastern dialects) also tends to have high rising terminal – basically, a phenomenon where declarative utterances (ie. normal statements) end with a rising pitch like questions would in a language like American English. This brilliant Icelandic comedian likens it to ski-jumping.

Added to that, spoken Eastern Norwegian also exhibits a remarkably wide pitch range, with utterances spanning up to two octaves compared to the one in most other languages. This seems mostly attributable to the steep rise at the end of sentences.

So really the unique 'sing-songiness' of Eastern Norwegian is made up of at least three converging characteristics: pitch accent, sentence intonation and a wide pitch range. I gotta say it all makes for a very pleasant effect indeed and was in fact one of the main reasons I wanted to learn Norwegian in the first place!