The boxes have been put onto around 17 streets in Hyde Park over the past few days as part of a trial.
Councillors decided to take action after complaints about people using residential streets as rat runs, as well as boy racers speeding.
Hyde Park and Headingley councillor Jonathan Pryor said: "Working with the highways department we put together some proposals that would tackle this dangerous and anti-social driving, and would also make it a lot safer for pedestrians walking though Hyde Park, and also take air pollution away from people’s homes.
"We know that inner city areas have some of the worst air pollution and bad health outcomes as a result so this was really important to tackle too.
"All streets are still completely accessible by car, but it essentially turns groups of housing into cul-de-sacs so dangerous driving is minimised."
Residents living in the streets were sent consultation documents eight weeks ago, and those who responded were "overwhelmingly positive".
The emergency services were consulted and still have full access to all streets.
But within days of the boxes appearing, several had been targeted by vandals and pushed over.
Coun Pryor said: "We posted on our Facebook page that this new trial traffic management scheme was now in place and the response has been really mixed.
"A lot of people are hugely positive and so pleased that they can feel safer and healthier in the homes, some people like the idea but want a few changes, and some people were against the scheme.
"Sadly a small minority became incredibly aggressive and abusive, and even threatened to vandalise the boxes.
"We then woke up to some criminal damage done to some of the plant boxes.
"This is unbelievably disappointing to see that just a small number of people think that their opinion is more important than everyone else’s, and further disappointing that they would vandalise things in our Hyde Park community."
Planters have been placed in Thornville Road, Pearson Avenue, Howden Place, Thornville Street, Alexandra Grove and the Norwoods, among others.
Coun Pryor said the damaged boxes would soon be fixed more securely to the street, while CCTV would be looked at "to ensure that people cannot continue to cause criminal damage without consequence".
He added: "We have equally had to remove a small number of people from our Facebook page who became abusive over the scheme, some we know do not even live in the area and were clearly angry that they can no longer rat run through.
"As politicians we are pretty thick skinned, but equally this is a scheme to make people safer, improve health and potentially save lives - so the behaviour of some is completely unacceptable."
The scheme is a trial, but Leeds City Council has said similar schemes are either planned or under consideration in Beeston, Chapeltown, Holbeck, Lincoln Green and Otley.
Coun Pryor added: "Anyone can let us know what they think of the scheme, positive or negative, we welcome all civil debate and want to know everyone’s thoughts."
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