Rumor sequences are generated recursively as follows: fix nonnegative integers b, c, and let z₀ = 0. For n ≥ 1, define z_n = (bz_n-1 + c) mod n, where the least nonnegative residue modulo n is taken. There have been a few papers dealing with the behavior of rumor sequences, but they all concern that behavior when the value of c is fixed. It turns out that if, for a given value of b, the rumor sequences for c = 0,1,2,... are written down, one below the other, some interesting and unexpected patterns appear in the columns of that array. These patterns are investigated, proving some, and, based on computer generated data, we make two conjectures.