They oppose polls tentatively scheduled for July, which the current ruling party would likely win.
Protest leader Suthep Thaugsuban told followers Friday night that the Senate should select a new prime minister, since the lower house has been dissolved since December.
A new Senate speaker sympathetic to the protesters' cause, Surachai Leangbunlertchai, was elected Friday by his colleagues.
Suthep called on Surachai to "Do whatever you need to submit the name of a new prime minister to the king immediately."
"This matter must end by Monday," Suthep warned, saying the protesters were ready to take over the offices of the prime minister themselves. "If it's not finished by then, we'll do it ourselves."
The protesters achieved a partial victory on Wednesday when the Constitutional Court ousted Prime Minister Yingluck Shinawatra, ruling that she had violated the constitution by transferring a senior civil servant to benefit her politically powerful family. Nine other Cabinet members were also forced from their posts.
Thailand's long-running political crisis began in 2006 when Yingluck's brother, former Prime Minister Thaksin Shinawatra, was toppled by a military coup after being accused of corruption, abuse of power and disrespect for King Bhumibol Adulyadej.
Thaksin, a former telecommunications billionaire, remains highly popular among the rural poor, and parties controlled by him have won every national election since 2001. The protesters, aligned with the opposition Democrat Party, say they want to remove all traces of his political machine from politics.
The protest group — backed mainly by Thailand's traditional elite, urban middle class and southerners — sent hundreds of people Friday to surround the main television stations. It said TV Channels 3, 5 and 7 had agreed to carry its leader's formal statements and not broadcast those of the Center for the Administration of Peace and Order, or CAPO, the government's security agency.
Spokesmen for Channels 3 and 5 could not be contacted, but a journalist in Channel 7's newsroom denied that the station had agreed to such terms, and said it would report the news normally.
Channel 11 or NBT, which is directly run by the government, reportedly moved its newsroom operations to a different location, anticipating interference from the protesters.
CAPO warned that media executives who agreed to the protesters' demands would face criminal charges.
The protesters also surrounded the compound housing the prime minister's office — mostly unused for several months because of the protests — and marched to Parliament.
There, Suthep presented a letter urging the Senate to take the lead in solving the country's crisis by setting up an interim government to reform politics before new polls are held. The lower house was dissolved in December, but February polls were disrupted by the protesters and invalidated by the Constitutional Court.
CAPO estimated the total number of protesters at just over 21,000.