Another consideration with rivers is prevailing wind and precipitation. Generally speaking weather systems will come off the ocean carrying large amounts of water. As the clouds begin to hit higher land they dump their water as either rain or snow. As a result, you will tend to get higher precipitation on the side nearest the prevailing winds, with a "rain shadow" forming a desert or semi-arid territory on the opposite side of your mountains. The extent of variation is mostly dependent on the height of the mountains.
One of the most extreme examples is the Andes in South America. These mountains are the longest and second highest mountain range on earth, and as such dramatically illustrate the effects of prevailing wind. In the north, the prevailing winds are the south-easterly trade winds. As a result, to the east of the Andes the land receives very high rainfall, and produces the Amazon rainforest. Meanwhile to the west, the land is in rain shadow, producing dry, arid conditions such as the Atacama Desert. However, to the south the prevailing winds reverse and come from the north-west. As a result in the lower portion of South America it is the eastern side that is arid, and the western coast receives abundant rainfall and has lush forests.
So when mapping, and considering rivers, it's important to determine where the most precipitation would occur, because that's the side that will have the most and largest rivers.