I just stumbled across an in depth discussion about the ins and outs of mowing your nature strip, and was interested in the fact that there seems to be a lot of confusion about it! Some people thought that it's the council's responsibility, others thought that it was the resident's responsibility, most agreed that not mowing it looks horrible, and those with corner blocks were unhappy that they had so much more to mow.
Walking around the streets of Melbourne, I've also seen some amusingly mown nature strips - the ones where one resident has mowed only up to the property line, leaving the rest of the nature strip wild and unruly in what can only be a passive aggressive statement to their neighbour. It seems like nature strips or verges are a confusing area and pit landlord against tenant, neighbour against neighbour and resident against council!
A lot of the cause for this trouble and strife is about the lack of consistency in the rules across councils, and therefore confusion with residents. So what are the rules? Wikipedia says that the residents mowing the nature strip is a 'custom', which implies that it's not a law. A few city council websites (Moreland, Whittlesea, etc) also support this - mowing your nature strip is not law but it is appreciated. The Moreland City Council website explicitly states "Council relies on the goodwill of residents to help us maintain nature strips.". In NSW (where nature strips are called verges) there are only a handful of councils who mow nature strips left (Marrickville, Botany Bay and Leichhardt).
When the council does mow it, people complain that it's not mown well - one comment was "the council's contractors are very good at turning grass into dirt, but not much else – they tend to cut the grass about half way through the roots". And given that one of the main purposes for nature strips and verges is to keep the dust down, this makes it a little pointless. But, as they'll only mow it 2-3 times a year they practically have to remove it so that the grass doesn't grow past the recommended 10 centimetres before the next mowing!
So, I'm afraid we can't answer the question for you but we can tell you to check with your local council to find out whether it's your job to mow the nature strip or verge. If it is your responsibility to mow the nature strip, you don't have to do it yourself - you can always find a quality gardener and lawn mower on Service Central. Remember that the nature strip is public property so public liability insurance is very important - and all Service Central businesses are appropriately insured.
To read a little more about the history or purpose of nature strips or verges, click here.