The style of playing that you described is the technique people use to learn to play from lead sheets (fake books) and is very common for playing non-classical (i.e. POP music). The fact that you have already learned to play a large variety of chords and can fluently read single line melody on the trebble cleff means that you are well on your way to some great keyboarding!!! If you played on an arranger keyboard instead of a Piano then you would already be making some amazing music but if the piano is your goal then your next step is to learn some rhythmic patterns for the left hand.
Think of your left hand being the bass player and your right hand being vocals. Obviously a Bass player could just plunk a single note every time there was a chord change but this would be boring. A good Bassist will play the notes of a chord using a rhythmic style suitable to the music that is being played. There are many different patterns, riffs and runs that can be learned (think in the hundreds) and they can be mixed and matched to produce truely musical accominaments to your right hand melody.
To get you started, here are some references:
"How to Play from a Fake Book" by Blake Neely. You already have these skills but it is a starting point for people new to this style of playing.
"The POP Piano Book" by Mark Harrison. This is the grandady of books for playing POP music. There are a number of different patterns included throughout the book but this volume has so much more to offer.
"100 ultimate Jazz Riffs", "100 Ultimate Blues Riffs for Piano/Keyboards" both by Andrew D. Gordon. These volumes include not only left hand patterns by right hand riffs and fill that can make your basic single note melodies really come alive.
"Mel Bay's Complete Book of Improvisation, Fills & Chord Progressions" by Gail Smith is an outstanding volume that will help with just about every aspect of your POP playing including left hand patterns.
John Axsom has collected a number of piano patterns that he sells on CD (PDF format) from his web site: http://users.wireweb.net/green/pianolessons.htm
"Pattern Piano and Keyboard" by David Sprunger is a video/book combination that teaches the basics of chord patterns to play accompinament to a vocalist. David has an amazing voice himself which makes the video lessons very interesting and inspiring. The book includes many patterns and he teaches how to put them together in a way that is very pleasing. His instructions assume no skills with reading music so is VERY good for the absolute beginner. http://www.PlayPianoTODAY.com/
This should get you going.