This is how I worked out the chords of the original movie scene. If you're playing the chords in Full-step-down tuning, they are played like Am, G and Em. (These easy chords sound quite good in Standard tuning as well, if you don't need the original key) Improvements are welcome!

^GmFar over the ^Fmisty mountains ^Gmcold To dungeons ^Fdeep and caverns ^Gmold We must a^Dmway ere break of ^Fday To find our ^Gmlong for^Dmgotten ^Gmgold

The pines were roaring ^Fon the ^Gmheight The winds were ^Fmoaning in the ^Dmnight The fire was red, it flaming ^Fspread The trees like ^Gmtorches ^Dmblazed with ^Gmlight

Howard Shore - Far over the misty mountains cold