28 ways

Let $a$, $b$, $c$ be the number on the spade, heart and diamond card chosen, respectively. If $a + c$ is not a multiple of $7$, let $k$ be the remainder obtained when $a + c$ is divided by $7$. Then $k$ satisfies $1 \le k \le 6$, and $a + b + c$ becomes a multiple of $7$ when and only when $b = 7 - k$. On the other hand if $a + c$ is a multiple of $7$, then $a + b + c$ cannot be a multiple of $7$, since $1 \le b \le 6$. Thus, the number of possible choices satisfying the requirement of the problem is the total number of choices for $(a, c)$, which equals $4 \times 8 = 32$, minus the number of cases for which $a + c$ is a multiple of $7$. The latter possibility occurs in 4 ways, namely, $(a, c) = (1, 6), (2, 5), (3, 4), (4, 3)$. Therefore, the answer we seek is $32 - 4 = 28$ ways.