As the Japanese "R" is the only liquid in the language, there is a considerable range of allophones used for it.

The first thing to remember is that the Japanese "R" sound is nothing at all like the English "R" sound. It is, in fact, often pronounced just like an English "L" sound. This is, in fact, how a kawaii anime girl is likely to pronounce it, especially when she is singing. If you are an English speaker and have trouble with foreign sounds, it is recommended that you use this pronunciation.

NOTE: This "L"-like pronunciation of "R" was deprecated by a Japanese person I wrote to. He said that it is used by younger people who have not been taught correctly.

If you wish to sound less like a kawaii anime girl, try a quick upward flap of the tongue, so that the tip of your tongue is just behind your front teeth. Do not roll the "R".

Starrynight writes: You might point out that the pronounciation can vary based on regional dialect. I'm fairly sure that some instances of Kansai dialect roll their rs.