Hi there gartok, this is the exact problem I once had, annoyed the pants off me!! But the solution (annoyingly) is fairly simple!
Basically, we need to get all the information from those checked boxes right? I'm assuming you've used the record ID as the value of each checkbox (but it don't really matter). Below is an example of a HTML tag for a checkbox taken from my own script
HTML Code:
<INPUT TYPE=CHECKBOX NAME="selitems[]" VALUE="14">
You should notice the strange square brackets after the name of the checkbox is set, selitems[] (selitems is just shortened from selecteditems, i know, super clever!).
What will happen now, is when the PHP form is submitted, it will place the value of everything that was ticked into the array, selitems[]. Then we need PHP to handle it!.
PHP Code:
foreach($selitems as $val) {
DeleteData("news_articles", "news_articles_id", "$val");
}
We use this loop to go through the $selitems array (note that if global arrays are not on this MUST be $_POST["selitems"]; or $_GET["selitems"]; whatever the send method was) and use it to delete the value in the DB as we want. Each value will be assigned to $val after each loop until it's gone through everything.
Oh yeah, the DeleteData() function won't work for you

cause that's a custom function, you just need to replace it with your own SQL Query Code.
I hope this helps you, as I was in a bit of a hurry to go out while posting this!