Invoke mysql with the -N (the alias for -N is --skip-column-names) option:
mysql -N ...
use testdb;
select * from names;
+------+-------+
| 1 | pete |
| 2 | john |
| 3 | mike |
+------+-------+
3 rows in set (0.00 sec)
Credit to ErichBSchulz for pointing out the -N alias.
To remove the grid (the vertical and horizontal lines) around the results use -s (--silent). Columns are separated with a TAB character.
mysql -s ...
use testdb;
select * from names;
id name
1 pete
2 john
3 mike
To output the data with no headers and no grid just use both -s and -N.
mysql -sN ...