You're stuck with your current table design.
You should always create tables with a unique index. One way to do this with the example you've shown, would be to add an additional column to your table to use as an object ID. You could call the column anything you want. In this example, it's called "oid", and looks like this in the CREATE TABLE command:
oid int unsigned not null auto_increment primary key
It's an integer column that auto increments itself, thus giving every row a unique identifier.
You will then be able to delete a row using:
DELTE FROM table_name WHERE oid=3 (or whatever number)