Just to offer my experience...
I used Access 2003 on my laptop computer for almost 3 months. I kept everything in Access 2000 format, in order to use it on my desktop as well, as I didn't put 2003 on my desktop.
I only ran into a couple of programming issues. One had to do with the HasData function. Can't remember the other one.
And, as MichaelRed suggests, there shouldn't be any issues between 2000 and 2003 as they are built on the same basic platform.
However, in due course I removed Access 2003 and went back to Access 2000. What drove me to removing 2003 was sudden and unexplainable program crashes which were always, without exception, unrecoverable. There is supposed to be a way to recover, through Office Recovery, but it surely never worked for me.
Recently, Microsoft has issued a service pack for Office 2003, and that, hopefully, might resolve some issues.
Tom