Ultimately if you have a substantial area of rhododendron that you are trying to get rid of you will need to use herbicide - it's the lesser of two evils. Glyphosate is relatively environmentally safe.
I cut the rhododendron down leaving a foot or so of stem, then I drill diagonally downwards into the side of the stem about 10cm deep (not the top!) with a 10-14mm drill bit in a battery operated drill. I then inject glyphosate based herbicide into the hole with a sheep dosing gun.
http://www.molevalleyfarmers.com/pd5_ANIM-MED_NEED[a]SYR_SYRINGES_05939_GUN-DRENCH-AUTOMATIC-20ML.htm
If you're not familiar with these a tube comes out of the back of the gun and conects to the bottom of a bottle you wear on your back. You can set the volume dispensed, then a single pull of the trigger will 3/4s fill the hole - no spillage, no measuring each time and very quick, easy single hand operation.
Once the rhododendron is cut, and the brash dealt with I just wander about with the drill in one hand and the dosing gun in the other - very easy.
(best done in summer)