In playing around with the left hand, here are some things to try:

• When switching to a new chord, play the note above it first. So when going into G you would play an A, when going into D you would play an E. I find that when I'm going to a higher bass note I want to lead into it from below and to a lower bass note I want to lead into it from above. This is leading into mostly from above: G B C E D A G (with chords).
• Play runs of consecutive notes that take you where you're trying to go. I especially like runs at the end of a phrase: downward D C B A G (with chords) or upward A B C Db D (with chords). These runs can be either using only notes in the scale or they can use all the notes not paying attention to whether they are in the scale. For example, the upward run above used a C which is not in the key of D. A similar run (A A B Db D) could use only notes in the key of D and has a bit of a different feel. Playing notes not in the key is called playing 'chromatically'.
• Play a few notes on the up beats. Previously we've only been playing left hand on the downbeats. It can sound good to play more frequently, as long as you don't overdo it. For example, I could play G GBC CDbD C B A G (with chords) adding in a couple up beats and then a run back down to G on the downbeats. You can also play runs with up beats, which gives you more places to put them in: G_D_GGABC_G_DCBAG (with chords). Runs played including up beats don't have the same power to say "and now we're starting something again" as the slower downbeat only ones, but they have an energy neverthless.

Here's an example of playing all this together over some random chords:

  A: |: G  C  G  D
        G  C  D  D :|
  B: |: Em C  D  D
        Em C  D  D :|
  

quick left hand example (It's kind of hard to play well without a tune, so I'm not that happy with this, but whatever).

Mostly you just want to try things and get so your hands can do what you tell them to. Have fun!

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