Counting is very much overrated as a tool for learning rhythm. There are many more reliable and far more life-filled models for regularity, symmetry, sequence, variation--all the things that are more rightly associated with rhythm than numbers. Let's leave the numbers for arithmetic.
Think instead of clapping patterns: A good game of Simon Says involving imitation of simple clapping patterns is more fun and potentially far more engaging than any amount of getting from 1 to 4 on time.
Or of moving an object, like a bow or a baton, to conduct a song or express a mood.
My best trick was getting children to draw shapes with both hands at the same time moving in contrary motion on a large sheet of paper. This is fun, and it is funny.
The hardest thing about beats is that they are too powerful in relation to tone quality. A child who is sensitive to tone quality cannot objectify time without losing contact with that more important, more elusive, and more deeply personal aspect of hearing.
