I think the current answers are all thinking of a very Earth like planet, including its orbit around its star.
As others have said, rivers have to flow somewhere, so your world would be more like full of seas or lakes than rivers.
How about we change things a little and have your planet orbit another, much more massive planet, such as a gas giant?
Get the giant in an orbit around a yellow star like our sun, and your planet will have as much sunlight as we do (though the gas giant may make for really interesting eclipses).
I am including the gas giant here because it will have tidal effects on the planet, much like the Moon does to Earth, but on a larger scale. Water will be pulled with extreme force towards the gas giant on the side that faces it, and away from the gas giant on the other side.
Now, if your planet is tidally locked, that water will be pretty much stuck. If not, your rivers will be pretty interesting, in that they will perpetually be running - but not necessarily towards the sea!
Huge edit @shufflepants has called my attention to the fact that the rivers wouldn't reverse their flow twice a day as I expected. So I am striking the previous text for this part of the answer, and including a couple scenarios that more probable, though still quite exciting.
In one scenario, your rivers could simple go around the world. On your planet, water will try to settle in the lowest places just like here on Earth. But there is no ocean, because there isn't a sea much below the river beds. Instead, the motive force behind the flow of the rivers are the tides. The tides will follow the gas giant, and the rivers will follow the tides. The interesting part is that this means the margins of the rivers will change within the course of a day. Some parts of the river may only be crossable by boat on the high tide, and may be crossable by foot on the lower tide.
Or, your planet could have oceans much like the earth. If it does, another interesting effect happens. You see, rivers end up in the oceans because usually they are more elevated than the sea. But during high tides, the water level on the sea may be higher, and the sea will go into rivers. When this happens on large rivers on Earth, the effects may be pretty... cinematographic, to say the least. In the Amazon river, the flow of the river is reversed in a five hundred miles strech starting where it meets the sea. Salt water enters the river in the form of waves that can be 12-feet tall at some points. This is what it looks like. You can surf on a single wave for more than half an hour!
This is technically called a tidal bore. There are dozens other places on Earth where this phenomenon happens. So you could just imagine that your world is an extrapolation of this, with these waves happening at least once a day for rivers that are to the east of an ocean (supposing your planet goes around the gas giant in the same way Earth goes around the Sun).
Both scenarios would give your planet interesting seasons. When the gas giant and the Sun rise together on the sky, or when the Sun and the gas giant are in opposite directions, the water of the high tide will be at its hottest ("tidal summer"?). But when they have a difference of 90°, you get the coldest high tide ("tidal winter"?)
As for geography of landmasses, it's up to you:
If the soil is very hard this movement of the rivers may act as a saw, digging ever deeper canyons on the land. So in high tides you get crossable rivers, but on the low tides you get deep ridges or fjords according to your worldbuilding taste.
If the soil is soft and will take the movement of the water, you can either have rolling hills or plains that get flooded in interesting patterns, or you can have places that look like tropical islands at high tide but look like deserts on the low tide.
You can also define how fast the water comes and goes... Depending on the force of the pull of the gas giant, the water may come slowly, or it may come like a tsunami (not very friendly to intelligent, surface dwelling life, though).
Also remember that your planet can have as many varied geographical features as real Earth does! You can mix all of the scenarios described above as you like.
One last thing, since I'm trying to be scientifically accurate here... Planets subjected to such tidal forces suffer from a special kind of geological effect because they are constantly being stretched in the directions towards and away from their "parent" planet. It is believed that this is what drives volcanism on Io (the most volcanically active place in the solar system). So your planet may be very active geologically too, if you want it to!