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What are good ways to approach playing over chords?

First, what kind of music are you playing? This greatly affects what's appropriate and what isn't.

In general, play the root on beat 1, and chordal tones on subsequent odd beats, nonchordal tones on even beats, with these even beats being slightly emphasized or even syncopated. Look for half or whole steps between chords when at all possible, half steps being preferable...Most descending fifth progressions make half-steps possible. This an extremely basic way to look at walking bass lines...feel free to deviate as much as you want.

Personally, when I play in a rock setting I usually approach things from a rhythmic standpoint first, and a tonal standpoint second. For instance, we're currently working on a song in 5/4 where the guitar plays a straight five groove with extremely quick harmonic motion (4 chords per measure). The drummer plays a syncopated five. I ride the middle by playing a line completely made up of root octaves with the upper octave being syncopated to match the drummer, but the lower octave thumping along on the straight five beats. I did some things with passing tones and other stuff in there, but ended up cutting all but the octaves out except for the last half beat of the measure where I throw in a passing half step between an AM chord and the bm chord starting the next measure. It took me some serious self-restraint to convince myself not to play with more harmony (the groove tonicizes b dorian...my favorite mode and tonic note...arrgh) but the bass line was better simpler in this case.

Wow...that got a little more in depth than I meant. To summarize...rhythm first, harmony second.
 
...rhythm first, harmony second.

+10

I mean, you can stay on the root note of the first chord and put a rhythm to it. You don't have to take up all the musical space with notes....rests count, too.

Here's a few link to previous TB threads that might be helpful:
http://www.talkbass.com/forum/showthread.php?p=80043 Creating bass lines
http://www.talkbass.com/forum/showthread.php?t=125536 Creating bass lines - Target Approach
http://www.talkbass.com/forum/showthread.php?t=491679 Playing behind/ahead of the beat
 
When I'm jamming, I usually garbage riff off 5ths and octaves until I can come up with a good line. While my hands are noodling around the 5th/octave "box", I'm trying to sing a bassline in my head, then I find it on the fretboard. I'm kind of an idiot savant (emphasis on "idiot") player, though- I know very little theory, I just come up with bass melodies and play them.


"Sometimes you have to play a long time to be able to play like yourself” -Miles Davis
 
One practical tip: I've been doing Gary Willis book "Fingerboard harmony for bass" for a few months now. On thing he emphasizes is avoiding root on beat one. When you are going from one chord to anoter, the point is not to choose the root but the nearest tone of the next chord. And then, little by little, you can carry on by adding passing tones and extensions.

I must say that I'm still in the very beginning of this process but every now and then I wonder is it really me who is playing!

So, I believe the book could be useful to you.