The class action lawsuit also targets the three companies' CEOs.
Mr Trump was suspended from his social accounts in January over public safety concerns in the wake of the Capitol riots, led by his supporters.
On Wednesday, Mr Trump called the lawsuit "a very beautiful development for our freedom of speech".
In a news conference from his golf resort in Bedminster, New Jersey, Mr Trump railed against social media companies and Democrats, who he accused of espousing misinformation.
"We are demanding an end to the shadow-banning, a stop to the silencing, and a stop to the blacklisting, banishing, and cancelling that you know so well," he said.
The suit requests a court order to end alleged censorship. Mr Trump added if they could ban a president, "they can do it to anyone".
None of the tech companies named have yet responded to the lawsuit, which was filed to a federal court in Florida.
Mr Trump was joined at the announcement by former Trump officials who have since created the not-for-profit America First Policy Institute.
The former president called the post that got him banned from Twitter, "the most loving sentence".
According to Twitter, the tweets that resulted in Mr Trump's ban for "glorification of violence" were from 8 January, two days after the rioting in the nation's capital. The riot followed his repeated claims, without evidence, that the election was rigged in Joe Biden's favour.
He wrote that the "great patriots" who voted for him will have "a giant voice" and "will not be disrespected or treated unfairly in any way, shape or form", and in another post said he would not attend President Joe Biden's inauguration.