Why do we wave the lulav forward by הודו, right by כי, back by טוב, left by כי, up by לעולם and down by חסדו?
If I understand your question correctly, you ask why these words specifically get these directions.
Heard from 2 local rabbanim - The assumption is that you begin with your face (front of your body) facing east or in shul, facing the Aaron. You always wave forward 1st and then wave going in a clockwise direction (i.e. - right is next) then back, left, finally up, then down.
The right side is more important than the left corresponding to the phrase "Y'mincha Hashem Ne'adari Bakoach" (Your right hand, G-d, is glorified (may be mistranslating a bit, here) with strength.)
Up precedes down, since we usually refer to the heavens as G-d's main abode.
Can't say, offhand, where the rabbanim got this info, but it does make some sense to me.
From Rabbi Reuven Hammer in Or Hadash commentary to Psalm 118:1-20:
The four lines following halleluyah are recited first by the leader of the service and then by the congregation. On Sukkot, when reciting the first two of these four lines, we take the lulav in the right hand and the etrog in the left, pitam (tip) facing upward, and wave them six times in the following order: forward, to the right, backward, to the left, upward, downward. This symbolizes the presence of God throughout the universe. When the Name of God is pronounced, we hold them straight and do not point in any specific direction, since that would imply that God is only in that spot.
Per my reading of the question, this answer doesn't really address why we shake the Lulav in each particular direction for each particular word.– LeeOct 17, 2014 at 5:23