How to make changes from Verse to Chorus more interesting: Chord progression help?

So I've noticed that a lot of my songs lately start out with a basic chord progression. When I decide it's time to change it up and maybe go to a second part, or to the chorus, it seems that I'm simply writing a different riff, often times using the same chords. Sometimes, they're in a different position on the neck, or a shortened "power chord" version of the same chord.

For instance, If my first chord used was a D chord, when I decide to write a second part to the song, I seem to start it again with another D chord. The riff or chord progression will be completely different after it, using different chords, different tempo, etc. Or, I'll write a metal riff that starts on E at the 7th fret on the A-string... then to change it, I simply move to the Open E string.... essentially, I'm using the same E chord.

Is this common? If not, can anyone give me a few suggestions on how to go about changing this up? Maybe provide a website or book I can pickup to help with this?

I had lessons for awhile, but my teacher wasn't helping much on this topic, and they were getting quite expensive. He had mentioned something once about a technique called a 5th or the 5th? or something like that... where you can figure out what the 5th is in your the Key you're using, then find out what a 5th away from that chord is, and you can use that chord as an effective alternate chord.... I don't know. I may have misunderstood.