We establish a quantitative result on norm convergence of triple ergodic averages with respect to three general commuting transformations by proving an r-variation estimate, r > 4, in the norm. We approach the problem via real harmonic analysis, using the recently developed techniques for bounding singular Brascamp-Lieb forms. It is not known whether such norm-variation estimates hold for all r>=2 as in the analogous cases for one or two commuting transformations, or whether such estimates hold for any r<infinity for more than three commuting transformations. This is joint work with Christoph Thiele and Lenka Slavikova.