Rumba walks are one of the things which when done well seem effortless but which in reality require a lot of practice and more multi tasking than you may have thought!
The technique is the same regardless of whether you are the leader or the follower and I will build it up step by step as I would if teaching it for the first time.
Because of the slow speed of the Rumba, one of the main things that you need to have to dance it well is control. I would recommend remaining within the rhythm of the Rumba when practising your walks and take two steps on the beat and then one step which holds the movement for two counts. (As you would if dancing..... counts 2, 3, 4-1.)
In a space where you can string quite a few steps together, begin by simply walking across the floor and back. Notice how you will normally walk with your heel hitting the floor first and often a relaxed centre. These are the first things that we are going to change.
The next time you walk, change the way that you hold yourself and lift out of your hips. Feel as though you have a big hook through your collar bone which is pulling you forward and slightly up.
After you get the feel of that, change the way that your feet contact the floor. Instead of stepping forward with a heel and allowing your feet to leave the floor, pretend that you have a large bill (£50 or $50 is usually sufficient) underneath the ball of each foot. The moment you lift your foot off the ground, you lose the money.
As you work with the "money" under your feet, think about the foot that you are bringing forward. Make sure it is extending out in front of you before you transfer your weight - pushing off the back foot, which then collects itself underneath you as you move your body weight forward. Stability will be helped if you turn out the foot you are stepping onto. (That means that you show the inside of the heel to the direction you are moving which points your toe diagonally out.)
Chances are, buy this point, you may have forgotten to think about your core and the fish hook through your collar bone. Whilst maintaining your footwork, try to feel the lift of your upper body out of your waist and hips. You should also attempt to keep your shoulders straight towards the direction you are walking and allow all of the work to be done below.... when done properly, this will be something you really feel in your stomach muscles! (It may help to put your arms out to the sides and then raise your hands 90 degrees to emphasise the necessity of keeping your upper body straight towards your LOD. You will be able to tell a lot easier if you are letting your shoulders move with your legs.)
So now, you have footwork that is pressing down into the floor (to keep your money under your toes) as your upper body lifts up and resists the natural movement that moving your legs causes in your shoulders. There should be a lot of resistance in your body as you walk - as though you are walking through treacle - and every single movement should be intentional and controlled.
Once you feel you have this under control, you can begin to work the hip movement into the walk. As you step forward onto the leg, your hip will move forward, to the side, and then settle slightly back - the way a lot of people naturally stand, with one hip pushed out slightly back. As you step forward onto the next foot, you repeat the process with the other hip. The total movement is a little like a figure 8. I will stress though, that this hip movement comes from the natural body movement and should not be forced. The more comfortable you are with the rest of the walk, the easier the hip movement will be to add.