Hajás does this by distinguishing instrumental gerunds
- that explain how an action was performed, such as "walking" in your sentence - and comitative gerunds
- which barely tell us that something was ongoing at the same time as something else, such as "whistling". Of course these terms have nothing to do with cases, but they seemed the most suitable and easy to remember to me.
Anyway, the sentence would be:
E sjend näzvelsel pëjnürj ej vímát, wolunug e qusjin lídamajnan.
/e ʃend nε'zveɫseɫ 'pøjnyr̝ ej 'viːmaːt 'wɔlunug e 'xuʃin 'liːdamajnan/
IND person INVERT.go.PAST walk.STRGER in.OPEN street.LOC whistle.COMGER IND cheerful.ACC music.DIMIN.ACC
A man came walking down the street, whistling a happy tune.
Note that the translation of "came" is not accurate. I wouldn't normally use any other verb than "walk" here, but if I hadn't used "come" as well this post would have been useless.