K-means is based on variance minimization. The sum-of-variance formula equals the sum of squared Euclidean distances, but the converse, for other distances, will not hold.
If you want to have a k-means like an algorithm for other distances (where the mean is not an appropriate estimator), use k-medoids (PAM). In contrast to k-means, k-medoids will converge with arbitrary distance functions!
For Manhattan distance, you can also use K-medians. The median is an appropriate estimator for L1 norms (the median minimizes the sum-of-differences; the mean minimizes the sum-of-squared-distances).
For your particular use case, you could also transform your data into 3D space, then use (squared) Euclidean distance and thus k-means. But your cluster centers will be somewhere underground!