The optimization process and recommendations:

1. Avoid OR Conditions By Using UNION (modified query below): In mosts cases, filtering using the OR operator cannot be applied using indexes. A more optimized alternative will be to split the query to two parts combined with a UNION clause, while each query holds one part of the original OR condition.
2. Create Optimal Indexes (modified query below): The recommended indexes are an integral part of this optimization effort and should be created before testing the execution duration of the optimized query.
3. Use UNION ALL instead of UNION (query line: 31): Always use UNION ALL unless you need to eliminate duplicate records. By using UNION ALL, you'll avoid the expensive distinct operation the database applies when using a UNION clause.

Optimal indexes for this query:

CREATE INDEX table_idx_col_z ON table (col_z);
CREATE INDEX table_idx_col_y ON table (col_y);
CREATE INDEX table_idx_col_x ON table (col_x);

The optimized query:

SELECT
        table_a,
        table_b,
        table_c 
    FROM
        ((SELECT
            table.a AS table_a,
            table.b AS table_b,
            table.c AS table_c 
        FROM
            table 
        WHERE
            table.col_z IN (
                ?, ?, ?
            )) 
    UNION
    (
        SELECT
            table.a AS table_a,
            table.b AS table_b,
            table.c AS table_c 
        FROM
            table 
        WHERE
            table.col_y IN (
                ?, ?, ?
            )
    ) 
UNION
(
    SELECT
        table.a AS table_a,
        table.b AS table_b,
        table.c AS table_c 
    FROM
        table 
    WHERE
        table.col_x IN (
            ?, ?, ?
        )
)
) AS union1