There are a few ways you can do this.
Remove everything up to and including the first _ in each filename:
Old name: [^_]+_(.+)
New name: \1
Type: Regular Expressions
Remove the first 3 characters from the start of a filename:
Old name: ...(.+)
New name: \1
Type: Regular Expressions
In the last example the number of dots before the first bracket define how many characters to remove, so if you want to remove 4 characters you'd use ....(.+) instead, and so on.
You have to use a rename script for that, unless you can think up a way to do your rename using a single expression (which is often hard/impossible).
There's an example VBScript script in the forums that you can modify to suit your needs so the only thing you have to learn is VBScript's slightly different regular expression syntax.