Okay, I am planning to make a planet that has three suns, and orbits one of them. Each of the suns has its own orbital path, so it's possible for the planet to have one sun, two suns, or three suns in its sky at a time. When there's one sun in the sky, the temperature is habitable, but when all three are in the sky, the day side becomes inhospitable. All orbits in the system are stable. What kind of orbits would the suns and the planet need for that to be scientifically sound?
Binary, with outlying star orbiting.
Your binaries are big and close. The star hosting your planet is at a distance. Algol is an example of such a system.
From the point of view of the Earth, Algol Aa1 and Algol Aa2 form an eclipsing binary because their orbital plane contains the line of sight to the Earth. The eclipsing binary pair is separated by only 0.062 astronomical units (au) from each other, whereas the third star in the system (Algol Ab) is at an average distance of 2.69 au from the pair...
The stars in a multi-star system can have a wide range of orbit size from less than 1 AU (the orbit radius of the Earth going around Sol) to light years. And the stars orbits around each other can be fairly circular to highly eccentric.
The planet's orbit can be very close to circular to highly eccentric.
It means you can pretty much dial up how much variability you want. Though you need to be concerned about the period between hot and cold if you want to be accurate. An eccentric orbit will mean the parts where it is far from its primary are longer because it moves slower in that part of the orbit. Remember Kepler's laws.
You can also make adjustments with whether the stars are orbiting as a tight group, as a single near a double, or as three widely spaced stars. Probably the last is hardest to predict the configuration over time.
Here is a fairly recent discovery of an exo planet. This picture is from that page.
The planet orbit is the red curve, and it orbits one of the stars, lets call it Alpha. The three stars orbit a common point in the middle. Two of the stars orbit each other, let's call them Beta and Gamma.
It's a little like the planet is Mercury, and then Beta and Gamma are Earth and the Moon. You can adjust this somewhat, as long as the planet does not get "stolen" by Beta and Gamma.
When the planet is to the left in the picture, it will see all three stars at the same time. It will get hot days, but it will also have night. When it's inside the star orbit, it will see Alpha during part of the day, and the other two during a different, overlapping, part of the day. And, since it's closer to the pair than when it is at the left, it will get more heat from them.
This is kind of backwards to what you want. When all three stars are on the same side, the planet is farthest from Beta and Gamma. So it gets net least heat. But it gets hot days followed by night, so possibly the most daily variation. When it is inside the blue circle it is closest to Beta and Gamma, so gets net most heat. But it will have atleast one star in the sky nearly the whole day, so there might be less daily variation.
Maybe no darkness the whole day is what you meant by all three stars being visble? In this case that would be when the planet got the most heat.
There are many possible arrangements of the orbits of the stars and the planet. For example, if the planet was orbiting well outside the orbit of the stars, it would be more like Mars being the planet, and Earth and the Moon still being Beta and Gamma. Then the planet would see one star for part of the day, then the other two would come into view. Depending on where it was in its orbit relative to Beta and Gamma, this would be more or less of a problem. When the planet was on the far side of Alpha relative to the other two, then they would provide less heat. And when on the same side, of course more. So the temperature on the planet would have a cycle that was determined by the interaction of its own year relative to the year of the Beta-Gamma pair.
There is a wide range of possible arrangments of the planet and the stars. You would need to adjust them so that you got as close to your desired temperature cycle as you could.