I do my share of Asatru meetup-style rune chanting. I particularly like chanting in small spaces, where hitting the right pitch will cause the rune's sound to vibrate in the air around you without sounding like it's coming from you at all.
What is called runesinging in the Finnish tradition is a different act altogether. The runesongs of the Finns are stories or ballads telling of how things were made, or why things exist as they are, or recounting wisdom. The term "runesongs" applies to both the stories that were adapted into the Kalevala, but also are the spells that Väinämöinen and others sing within the stories to shape their magic.
The power of the runesong, then, is in the storytelling and the intent of the storyteller. While courting the Rainbow Maiden, Väinämöinen injured himself with an iron axe. To heal the injury magically, he had to learn the origin of iron. Then he had to work the spell itself.
As I practice it, runesinging has the same two-part process. First is the matter of understanding what's being asked for. This doesn't necessarily mean understanding it with SCIENCE - an idea of the mythic context works too. The important thing is to know what you want - what you're asking for, and how likely it is, and how it's likely to come about.
When it comes to learning songs, I recommend you start by reading at least the Kalevala in translation. If you're familiar with any ballads that were originally from an oral tradition, you'll notice certain ways in which meter and repetition weigh into the rune. Gods and heroes who are called upon are named repeatedly, with multiple epithets. You want to make sure you're getting the right person, after all.
Where else do you find incredibly repetitive, structured, rhyming songs? Oh, right, pop music. I'm of the opinion that you can use pretty much any song for runesinging as long as it tells an appropriate story and as long as you have the knowledge of the origin and the will to back it up. (Elias Lönnrot I ain't, sorry.) The important thing is that the song has the right associations and invokes the right emotions/mindset for you. Hell, I used a Hanson song once. It worked, that's what matters.
So it comes back to the two parts: intent, which takes the form of learning the origin of the thing you want to bring about - the lore of it, the way to get there, whatever it takes to understand that thing, and will, which takes the form of the song itself, sung with passion and strength behind it, energy streaming from it.
Works for me, anyway.