Stop thinking in chords
Yea b00vie, what’s with that? Thinking in chords is like wanting to use a hammer for fixing an electronic circuit (sure, it works in some extreme cases). Though I remember when I was just a little bit younger, and had my crappy acoustic guitar, and I have learned to strum out the chords of quite an amount of tunes. And man, I felt like I knew music inside out. Truth is, it was just a tiny part of music. Let’s see the other part.
Strum it like you mean it
Okay, so there’s really nothing wrong in taking your guitar or sit at the piano and play those chords as an accompaniment while you sing. It’s sort of like laying down a carpet under the main attraction, which is your voice: the melody. But if you tend to listen to all kinds of different music, you’ll notice that the more interesting ones (most of those you tend to not get bored of as quickly) have more than that. They are more than a textural, chordal rug under a single melody line. Be it a classical piece or some riffing rock tune, there will be simultaneous melody lines playing together: counter-melodies. The fun part is, they do form certain chords (or at least double stops with the main line), but they are their own entity. They usually have a bass line as well, forming another melody line down there (not there, get out of the gutter).
More melodies for the poor
But why is it all that good and special? Well, my friend, if you have ever felt that good music moved you, I have good news. Melody emphasizes exactly that: movement. Think about it a bit. If you keep strumming or hitting block chords, the single tones or voices of those chords move only at the strict tempo you’re playing them with, and they move all at the same time, in sync. In some extreme cases, the rhythm of such chording is identical with the rhythm of the melody line. Meh.
Movement is king
Even just arpeggiating the chords generates certain little melodies, which creates movement. And you can do even better, with actively thinking up counter-melodies for your songs. It might be a lot harder to realize if you are a solo player (think of all those Bach pieces), but just keep at it. It’s worth the effort.