Instead of settling for chords that simply sit there and create harmony, break them up and create fascinating and robust melodic lines. Try breaking up the chords into sixteenth notes and create arpeggios with the music. Another option is to insert pitches between the chord tones to create progressions that really move.
Most people are familiar with chord tones built upon thirds, but too many composers are afraid to introduce chord tones that do not fit with a traditional major or minor triad. Try adding sevenths, ninths and thirteenths to your basic chord progressions and see whether you like what you hear. A good rule of thumb is to use proper voice leading by using mostly steps and avoiding skips or leaps to resolve these chords. By doing so, you will prevent music that sounds random and disorganized.
If you are growing tired of creating triad-based chords, try moving to a quartal system. Aaron Copland was famous for his quartal harmonies that evoke a spacious and open harmonic backdrop. Build your bass line as you normally would and then simply create intervals in fourths from the bass line. Try to experiment with different voicing of the intervals to get the right sound. You can even condense the fourths into smaller areas to create clusters that will provide your music with an edgy and distinctive sound.
Counterpoint is a great subject to study in order to understand how chord progressions can be formed. Studying 16th- and 18th-century counterpoint will help you to develop your technique and learn the mechanics of chord progressions. By learning first through fifth species counterpoint, a world of options will begin to open up for you.
First species three-part counterpoint will teach you how to create chords that work as a cohesive whole. It is possible to learn how to create progressions that actually work with and enhance a melody rather than simply support it. By learning counterpoint, you can create music that merges in a synergistic way with the melody.
Counterpoint is a subject that is too complex to address in a single article, but there are online courses, community colleges and universities where you can study the subject. The Resource section includes a website that can teach you about counterpoint.